Part 7

Determinants and Quadratic Forms

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Section 7.1Evaluation of Determinants. Solution of Linear Equations[20/100%]

Problem 7.1

Let the $n$ vertices of a polyhedron be numbered in a definite order. Define a determinant of $n$ th order as follows:

If the $\lambda$ th and the $\mu$ th vertices are the two end-points of an edge of the polyhedron, let $a_{\lambda \mu } = a_{\mu \lambda } = 1$ .

If the line joining the $\lambda$ th and the $\mu$ th vertices is not an edge of the polyhedron, let $a_{\lambda \mu } = 0$ . In particular $a_{\lambda \lambda } = 0$ , $\lambda = 1, 2, \ldots , n$ .

Show that the value of this determinant is independent of the manner of numbering the vertices. Form and calculate the determinants for the regular tetrahedron, hexahedron and octahedron.

Problem 7.1.1

(continued). If exactly $p$ edges end in each vertex the determinant is divisible by $p$ .

Problem 7.1.1

(continued). If exactly $p$ edges end in each vertex the determinant is divisible by $p$ .

Problem 7.2

Calculate

\begin{vmatrix} 1 & 1 & 1 & \ldots & 1 \\ b_{1} & a_{1} & a_{1} & \ldots & a_{1} \\ b_{1} & b_{2} & a_{2} & \ldots & a_{2} \\ \multicolumn {5}{c}{\dotfill } \\ b_{1} & b_{2} & b_{3} & \ldots & a_{n} \end{vmatrix}

Problem 7.3

Prove the identity:

\left|\frac{1}{a_{\lambda } + b_{\mu }}\right|_{1}^{n} = \frac{\prod _{j > k}^{1, 2, \ldots , n} (a_{j} - a_{k})(b_{j} - b_{k})}{\prod _{\lambda , \mu }^{1, 2, \ldots , n} (a_{\lambda } + b_{\mu })}.

Problem 7.4

Denoting the determinant of the quadratic form

\sum _{\lambda =1}^{n} \sum _{\mu =1}^{n} \frac{x_{\lambda } x_{\mu }}{\lambda + \mu }

by $D_{n}$ , we have

D_{n} = \frac{[1! 2! \ldots (n-1)!]^{3} n!}{(n+1)! (n+2)! \ldots (2n)!}.

Also calculate the determinant $D_{n}(\alpha )$ of the quadratic form

\sum _{\lambda =1}^{n} \sum _{\mu =1}^{n} \frac{x_{\lambda } x_{\mu }}{\lambda + \mu + \alpha }, \qquad \alpha > -2.

Problem 7.5

\left|(a_{\lambda } - b_{\mu })^{n-1}\right|_{1}^{n} = \prod _{\nu =1}^{n-1} (n-\nu )^{n-2\nu } \prod _{j > k}^{1, 2, \ldots , n} (a_{j} - a_{k})(b_{j} - b_{k}).

Problem 7.6

Derive Theorem 86 of Part V from Theorem 48 of Part V by means of a transformation of the determinant $\left|F(\alpha_{\lambda } \beta_{\mu })\right|$ .

Problem 7.7

Setting $f(x) = (r_{1} - x)(r_{2} - x) \ldots (r_{n} - x)$ , we have

\begin{vmatrix} r_{1} & a & a & \ldots & a \\ b & r_{2} & a & \ldots & a \\ b & b & r_{3} & \ldots & a \\ \multicolumn {5}{c}{\dotfill } \\ b & b & b & \ldots & r_{n} \end{vmatrix} = \frac{a f(b) - b f(a)}{a - b}.

Remark.

Add to all the $n^{2}$ elements the variable $x$ . The determinant formed in this way is a linear function of $x$ and as such can be determined from two particular values.

Problem 7.8

Set $\Delta = ad - bc$ . Then the functional determinant

\frac{\partial (a\Delta , b\Delta , c\Delta , d\Delta )}{\partial (a, b, c, d)} = 3\Delta ^{4}.

Problem 7.9

Prove that $\rho - 2$ is a divisor of the expression

\begin{vmatrix} \rho & \dfrac {l}{m} + \dfrac {m}{l} & \dfrac {n}{l} + \dfrac {l}{n} \\ ]1ex] \dfrac {l}{m} + \dfrac {m}{l} & \rho & \dfrac {m}{n} + \dfrac {n}{m} \\[1ex] \dfrac {n}{l} + \dfrac {l}{n} & \dfrac {m}{n} + \dfrac {n}{m} & \rho \end{vmatrix}, \qquad l \neq 0, \; m \neq 0, \; n \neq 0.

Determine the remaining factors.

Problem 7.10

The determinant

\left|1, \; x_{\nu }, \; x_{\nu }^{2}, \; \ldots , \; x_{\nu }^{n-q-1}, \; x_{\nu }^{n-q+1}, \; \ldots , \; x_{\nu }^{n-1}, \; x_{\nu }^{n}\right|, \qquad \nu = 1, 2, \ldots , n

is an alternating rational entire function of the $n$ numbers $x_{1}, x_{2}, \ldots , x_{n}$ and thus divisible by the product of differences

\prod _{j > k}^{1, 2, \ldots , n} (x_{j} - x_{k}).

Show that the quotient $S_{q}$ is equal to the $q$ th elementary symmetric function of the $n$ numbers $x_{1}, x_{2}, \ldots , x_{n}$ .

Problem 7.11

Let the numbers $a_{0}, a_{1}, a_{2}, \ldots , a_{n}$ be different from $0$ . Then we have identically in $z$

a_{0} \begin{vmatrix} z + \dfrac {a_{1}}{a_{0}} & \dfrac {a_{2}}{a_{1}} & \dfrac {a_{3}}{a_{2}} & \ldots & \dfrac {a_{n-1}}{a_{n-2}} & \dfrac {a_{n}}{a_{n-1}} \\ ]1ex] -\dfrac {a_{1}}{a_{0}} & z & 0 & \ldots & 0 & 0 \\[1ex] 0 & -\dfrac {a_{2}}{a_{1}} & z & \ldots & 0 & 0 \\ \multicolumn {6}{c}{\dotfill } \\ 0 & 0 & 0 & \ldots & -\dfrac {a_{n-1}}{a_{n-2}} & z \end{vmatrix} = a_{0} z^{n} + a_{1} z^{n-1} + \cdots + a_{n}.

Problem 7.11.1

Let $a_{0}, a_{1}, a_{2}, \ldots , b_{0}, b_{1}, b_{2}, \ldots$ denote constants, $z$ a variable. Suppose that $a_{0} \neq 0$ and that we have identically

(a_{0} + a_{1} z + a_{2} z^{2} + \cdots )(b_{0} + b_{1} z + b_{2} z^{2} + \cdots ) = 1.

Then

a_{0}^{n+1} b_{n} = (-1)^{n} \begin{vmatrix} a_{1} & a_{0} & 0 & 0 & \ldots & 0 \\ a_{2} & a_{1} & a_{0} & 0 & \ldots & 0 \\ a_{3} & a_{2} & a_{1} & a_{0} & \ldots & 0 \\ \multicolumn {6}{c}{\dotfill } \\ a_{n} & a_{n-1} & a_{n-2} & a_{n-3} & \ldots & a_{1} \end{vmatrix}.

(This serves as a method of computing the reciprocal of a given power series. The supposed identity may be merely formal; no considerations of convergence are involved.)

Problem 7.11.1

Let $a_{0}, a_{1}, a_{2}, \ldots , b_{0}, b_{1}, b_{2}, \ldots$ denote constants, $z$ a variable. Suppose that $a_{0} \neq 0$ and that we have identically

(a_{0} + a_{1} z + a_{2} z^{2} + \cdots )(b_{0} + b_{1} z + b_{2} z^{2} + \cdots ) = 1.

Then

a_{0}^{n+1} b_{n} = (-1)^{n} \begin{vmatrix} a_{1} & a_{0} & 0 & 0 & \ldots & 0 \\ a_{2} & a_{1} & a_{0} & 0 & \ldots & 0 \\ a_{3} & a_{2} & a_{1} & a_{0} & \ldots & 0 \\ \multicolumn {6}{c}{\dotfill } \\ a_{n} & a_{n-1} & a_{n-2} & a_{n-3} & \ldots & a_{1} \end{vmatrix}.

(This serves as a method of computing the reciprocal of a given power series. The supposed identity may be merely formal; no considerations of convergence are involved.)

Problem 7.11.2

Compute the coefficient of $z^{n}$ in the expansion of $[(1+z)^{a} - xz]^{-1}$ first from Problem 11.1 then directly.

Problem 7.11.2

Compute the coefficient of $z^{n}$ in the expansion of $[(1+z)^{a} - xz]^{-1}$ first from Problem 11.1 then directly.

Problem 7.11.3

Analogously to Problem 11.2, compute the coefficient of $z^{n}$ in the expansion of $[e^{z} - xz]^{-1}$ .

Problem 7.11.3

Analogously to Problem 11.2, compute the coefficient of $z^{n}$ in the expansion of $[e^{z} - xz]^{-1}$ .

Problem 7.12

Let $a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}, c$ be real numbers. The system

\begin{aligned} -a_{3} x_{2} + a_{2} x_{3} & = b_{1}, \\ a_{3} x_{1} \phantom{{} + a_{2} x_{2}} - a_{1} x_{3} & = b_{2}, \\ -a_{2} x_{1} + a_{1} x_{2} \phantom{{} - a_{1} x_{3}} & = b_{3}, \\ a_{1} x_{1} + a_{2} x_{2} + a_{3} x_{3} & = c \end{aligned}

can be consistent in two different ways. In one case the unknowns $x_{1}, x_{2}, x_{3}$ are completely undetermined, in the other case they are completely determined.

Problem 7.13

Let the entire function

1 + c_{1} z + c_{2} z^{2} + c_{3} z^{3} + \cdots

have all distinct zeros $a_{1}, a_{2}, \ldots , a_{n}, \ldots ,$

0 < \left|a_{1}\right| < \left|a_{2}\right| < \cdots < \left|a_{n}\right| < \cdots .

Consider the system of $n$ equations

\begin{aligned} 1 + a_{1} u_{1}^{(n)} + a_{1}^{2} u_{2}^{(n)} + \cdots + a_{1}^{n} u_{n}^{(n)} & = 0, \\ 1 + a_{2} u_{1}^{(n)} + a_{2}^{2} u_{2}^{(n)} + \cdots + a_{2}^{n} u_{n}^{(n)} & = 0, \\ & \dotfill \\ 1 + a_{n} u_{1}^{(n)} + a_{n}^{2} u_{2}^{(n)} + \cdots + a_{n}^{n} u_{n}^{(n)} & = 0 \end{aligned}

that completely determine the $u_{k}^{(n)}$ . If

\frac{1}{\left|a_{1}\right|} + \frac{1}{\left|a_{2}\right|} + \cdots + \frac{1}{\left|a_{n}\right|} + \cdots

is convergent, then $\lim_{n \rightarrow \infty } u_{k}^{(n)}$ exists, but it is not necessarily true that

\lim _{n \rightarrow \infty } u_{k}^{(n)} = c_{k}.

Remark.

One solution of the infinite system

1 + a_{\nu } u_{1} + a_{\nu }^{2} u_{2} + \cdots = 0, \qquad \nu = 1, 2, 3, \ldots

is: $u_{1} = c_{1}, u_{2} = c_{2}, u_{3} = c_{3}, \ldots$ .

Problem 7.14

In the equations

\begin{aligned} c_{11} z_{1} + c_{12} z_{2} + \cdots + c_{1n} z_{n} & = 0, \\ c_{21} z_{1} + c_{22} z_{2} + \cdots + c_{2n} z_{n} & = 0, \\ & \dotfill \\ c_{n1} z_{1} + c_{n2} z_{2} + \cdots + c_{nn} z_{n} & = 0 \end{aligned}

let the coefficients and the unknowns be complex:

c_{\lambda \mu } = a_{\lambda \mu } + i b_{\lambda \mu }, \qquad z_{\mu } = x_{\mu } + i y_{\mu },

$a_{\lambda \mu }, b_{\lambda \mu }, x_{\mu }, y_{\mu }$ real. For the equations to admit not only the identically zero solution

z_{1} = z_{2} = \cdots = z_{n} = 0, \qquad \text{i.e.} \quad x_{1} = x_{2} = \cdots = x_{n} = y_{1} = y_{2} = \cdots = y_{n} = 0,

it is necessary and sufficient that the determinant $\left|c_{\lambda \mu }\right|_{1}^{n}$ vanishes. This yields two equations in the $2n^{2}$ real numbers $a_{\lambda \mu }, b_{\lambda \mu }$ . On the other hand the given equations may be written as $2n$ linear homogeneous equations for $2n$ real unknowns. The necessary and sufficient condition for the existence of a not identically vanishing solution now consists in the vanishing of a real determinant, i.e. in one equation in the $a_{\lambda \mu }, b_{\lambda \mu }$ . How can this be correct?

Problem 7.15

The six terms in the expansion of a third-order determinant cannot all be positive.

Problem 7.16

The rule for the expansion of a determinant consists of two parts. The first part specifies what products are to be formed from the elements, while the second part determines the sign of these products.

The second part may be simplified in the case of second-order determinants in the following manner:

Associate with the elements

\begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} \text{ the signs } \begin{pmatrix} + & + \\ - & + \end{pmatrix} \text{ respectively.}

Then form the products prescribed in the first part from the elements, each with its fixed associated sign.

Now prove that a corresponding simplification is impossible in the case of a determinant of order higher than two, i.e. it is impossible to associate with the $n^{2}$ elements $n^{2}$ fixed signs in such a way that, if we form the products prescribed in the first part of the expansion rule from the elements each with its associated sign, we automatically obtain the correct sign in all the products.

Section 7.2Power Series Expansion of Rational Functions[18/100%]

Problem 7.17

Let the power series $a_{0} + a_{1} z + a_{2} z^{2} + \cdots$ represent a rational function whose denominator is of precise degree $q$ and whose numerator is of precise degree $p-1$ . Assume that the numerator and the denominator have no common divisor. Setting $d = \max (0, p-q)$ we have

A_{d}^{(q+1)} = A_{d+1}^{(q+1)} = A_{d+2}^{(q+1)} = \cdots = 0.

Remark.

In Problems 17--34 we consider the Hankel (or recurrent) determinants

\begin{vmatrix} a_{n} & a_{n+1} & a_{n+2} & \ldots & a_{n+r-1} \\ a_{n+1} & a_{n+2} & a_{n+3} & \ldots & a_{n+r} \\ a_{n+2} & a_{n+3} & a_{n+4} & \ldots & a_{n+r+1} \\ \multicolumn {5}{c}{\dotfill } \\ a_{n+r-1} & a_{n+r} & a_{n+r+1} & \ldots & a_{n+2r-2} \end{vmatrix} = A_{n}^{(r)},

formed from the coefficients of the power series $a_{0} + a_{1} z + a_{2} z^{2} + \cdots + a_{n} z^{n} + \cdots$ ; $a_{n}$ is the leading entry, $r$ is the number of rows.

Problem 7.18

Let $d, q$ be non-negative integers. If we have

A_{d}^{(q)} \neq 0, \quad A_{d+1}^{(q)} \neq 0, \quad A_{d+2}^{(q)} \neq 0, \quad A_{d+3}^{(q)} \neq 0, \quad \ldots

A_{d}^{(q+1)} = 0, \quad A_{d+1}^{(q+1)} = 0, \quad A_{d+2}^{(q+1)} = 0, \quad \ldots

then the power series may be represented as the quotient of two polynomials, where the denominator is of degree $q$ and the numerator is of degree $\leq q+d-1$ .

Remark.

Investigate the dependence of the linear forms

L_{n}(x) = a_{n} x_{0} + a_{n+1} x_{1} + a_{n+2} x_{2} + \cdots + a_{n+q} x_{q}.

Problem 7.19

A_{n}^{(r)} A_{n+2}^{(r)} - A_{n}^{(r+1)} A_{n+2}^{(r-1)} = (A_{n+1}^{(r)})^{2}.

Problem 7.20

If we have

A_{m}^{(q+1)} = A_{m+1}^{(q+1)} = A_{m+2}^{(q+1)} = A_{m+3}^{(q+1)} = \cdots = A_{m+t-1}^{(q+1)} = 0,

then the $t$ determinants $A_{m+1}^{(q)}, A_{m+2}^{(q)}, A_{m+3}^{(q)}, \ldots , A_{m+t}^{(q)}$ are either all $=0$ , or all $\neq 0$ .

Problem 7.21

The triangular array

\begin{array}{ccccccccc} A_{0}^{(1)} & A_{1}^{(1)} & A_{2}^{(1)} & A_{3}^{(1)} & A_{4}^{(1)} & * & * & * & * \\ ]0.5ex] & A_{0}^{(2)} & A_{1}^{(2)} & A_{2}^{(2)} & * & * & * & * & * \\[0.5ex]& & A_{0}^{(3)} & * & * & * & A_{n+2}^{(r-1)} & * & * \\ ]0.5ex] & & & * & * & A_{n}^{(r)} & A_{n+1}^{(r)} & A_{n+2}^{(r)} & * \\[0.5ex]& & & & * & * & A_{n}^{(r+1)} & * & * \end{array}

is arranged so that a right angle opening upwards and bisected by the vertical contains only minors of the determinant standing at the vertex of the angle.

Problem 7.22

(1)(2)(3)

In the array of Problem 21, $A_{n}^{(r)}$ and $A_{n+1}^{(r)}$ constitute a horizontal pair, $A_{n+1}^{(r)}$ and $A_{n}^{(r+1)}$ constitute a vertical pair, $A_{n}^{(r)}$ and $A_{n+1}^{(r+1)}$ constitute a diagonal pair, $A_{n+1}^{(r)}$ and $A_{n-1}^{(r+1)}$ also form a diagonal pair, cross-wise to $A_{n}^{(r)}$ and $A_{n+1}^{(r+1)}$ .

(1)

If a diagonal pair vanishes, then the diagonal pair cross-wise to it also vanishes.

(2)

If a horizontal pair vanishes, then either the horizontal pair lying above it or the one lying below it also vanishes.

(3)

If a vertical pair vanishes, then either the neighbouring vertical pair on the left or the one on the right also vanishes.

Theorems (1), (2), (3) also hold on the slanting boundary line of the array of Problem 21.

Problem 7.23

If of the infinitely many determinants

A_{0}^{(k+1)}, \quad A_{1}^{(k+1)}, \quad A_{2}^{(k+1)}, \quad \ldots , \quad A_{n}^{(k+1)}, \quad \ldots

only finitely many are different from $0$ , then the power series $a_{0} + a_{1} z + a_{2} z^{2} + \cdots + a_{n} z^{n} + \cdots$ represents a rational function whose denominator is of degree not greater than $k$ . [Problems 20, 18.]

Problem 7.24

If of the infinitely many determinants

A_{0}^{(1)}, \quad A_{0}^{(2)}, \quad A_{0}^{(3)}, \quad \ldots , \quad A_{0}^{(n)}, \quad \ldots

only finitely many are different from $0$ , then the power series $a_{0} + a_{1} z + a_{2} z^{2} + \cdots + a_{n} z^{n} + \cdots$ represents a rational function.

Problem 7.25

If of the infinitely many determinants

A_{0}^{(1)}, \quad A_{1}^{(2)}, \quad A_{2}^{(3)}, \quad A_{3}^{(4)}, \quad \ldots , \quad A_{n}^{(n+1)}, \quad \ldots

A_{1}^{(1)}, \quad A_{2}^{(2)}, \quad A_{3}^{(3)}, \quad \ldots , \quad A_{n}^{(n)}, \quad \ldots

only finitely many are different from $0$ , then the power series $a_{0} + a_{1} z + a_{2} z^{2} + \cdots + a_{n} z^{n} + \cdots$ represents a rational function.

Problem 7.26

Show that the sufficient criteria given in Problems 23, 24, 25 for the power series $a_{0} + a_{1} z + a_{2} z^{2} + \cdots + a_{n} z^{n} + \cdots$ to represent a rational function are also necessary.

Problem 7.27

The rank of the matrix $\mathfrak {H}$ is finite if and only if the power series $a_{0} + a_{1} z + a_{2} z^{2} + \cdots + a_{n} z^{n} + \cdots$ represents a rational function.

Remark.

An infinite matrix

\begin{pmatrix} a_{00} & a_{01} & a_{02} & \ldots \\ a_{10} & a_{11} & a_{12} & \ldots \\ \multicolumn {4}{c}{\dotfill } \end{pmatrix}

is said to be of finite rank $r$ if all determinants of order $r+1$ contained in it vanish but not all those of order $r$ .

In the infinite Hankel matrix $\mathfrak {H}$

\mathfrak {H} = \begin{pmatrix} a_{0} & a_{1} & a_{2} & \ldots \\ a_{1} & a_{2} & a_{3} & \ldots \\ a_{2} & a_{3} & a_{4} & \ldots \\ \multicolumn {4}{c}{\dotfill } \end{pmatrix}

( $a_{\lambda \mu } = a_{\lambda +\mu }$ ) we term the rank just defined also the gross rank of $\mathfrak {H}$ . By striking out the first $k$ columns (or rows) we obtain from $\mathfrak {H}$ the matrix $\mathfrak {H}_{k}$

\mathfrak {H}_{k} = \begin{pmatrix} a_{k} & a_{k+1} & a_{k+2} & \ldots \\ a_{k+1} & a_{k+2} & a_{k+3} & \ldots \\ a_{k+2} & a_{k+3} & a_{k+4} & \ldots \\ \multicolumn {4}{c}{\dotfill } \end{pmatrix}

whose rank we shall denote by $r_{k}$ . We have $\mathfrak {H}_{0} = \mathfrak {H}$ , $r_{0} = r$ , $r_{0} \geq r_{1} \geq r_{2} \geq \cdots$ . The minimum of the sequence of numbers $r_{0}, r_{1}, r_{2}, \ldots$ , which is necessarily attained after finitely many steps, we term the net rank of $\mathfrak {H}$ .

Problem 7.28

The gross rank of $\mathfrak {H}$ is equal to the order of the last non-vanishing determinant in the infinite sequence

A_{0}^{(1)}, \quad A_{0}^{(2)}, \quad A_{0}^{(3)}, \quad A_{0}^{(4)}, \quad \ldots .

If this is $A_{0}^{(p)}$ then the net rank is equal to the order of the first non-vanishing determinant in the finite sequence

A_{1}^{(p)}, \quad A_{2}^{(p-1)}, \quad A_{3}^{(p-2)}, \quad \ldots , \quad A_{p}^{(1)}.

(The net rank $=0$ if these $p$ determinants all vanish.)

Problem 7.29

The net rank of $\mathfrak {H}$ (assumed finite) is equal to the degree of the denominator of the rational function represented by $a_{0} + a_{1} z + a_{2} z^{2} + \cdots + a_{n} z^{n} + \cdots$ . The rational function is a proper fraction if and only if the gross rank is equal to the net rank. If the gross rank is greater than the net rank, then the gross rank exceeds the degree of the numerator by one. (The numerator and the denominator of the rational function are assumed to have no common divisors; the degree here is the precise degree.)

Problem 7.30

The power series

a_{0} + \frac{a_{1} z}{1!} + \frac{a_{2} z^{2}}{2!} + \cdots + \frac{a_{n} z^{n}}{n!} + \cdots

satisfies a linear homogeneous differential equation with constant coefficients if and only if the determinants

A_{0}^{(1)}, \quad A_{0}^{(2)}, \quad A_{0}^{(3)}, \quad \ldots , \quad A_{0}^{(n)}, \quad \ldots ,

apart from a finite number, all vanish.

Problem 7.31

Let $Q_{n}(z)$ be a polynomial of degree $n$ , $Q_{n}(0) = 1$ , $n = 0, 1, 2, \ldots$ . For brevity set

a_{0} + a_{1} z + a_{2} z^{2} + \cdots = f(z), \qquad Q_{k}(z) f(z) = \square _{k} a_{0} + \square _{k} a_{1} z + \square _{k} a_{2} z^{2} + \cdots ,

Q_{k}(z) Q_{l}(z) f(z) = \square _{k} \square _{l} a_{0} + \square _{k} \square _{l} a_{1} z + \square _{k} \square _{l} a_{2} z^{2} + \cdots

( $\square_{k} \square_{l} a_{n}$ is a homogeneous linear expression in $a_{n}, a_{n-1}, a_{n-2}, \ldots , a_{n-k-l}$ ).

We have

A_{0}^{(n+1)} = \begin{vmatrix} a_{0} & \square _{1} a_{1} & \ldots & \square _{n} a_{n} \\ \square _{1} a_{1} & \square _{1} \square _{1} a_{2} & \ldots & \square _{1} \square _{n} a_{n+1} \\ \multicolumn {4}{c}{\dotfill } \\ \square _{n} a_{n} & \square _{n} \square _{1} a_{n+1} & \ldots & \square _{n} \square _{n} a_{2n} \end{vmatrix}.

Problem 7.32

Let $a_{0}/z + a_{1}/z^{2} + a_{2}/z^{3} + \cdots$ be the power series expansion of a rational function whose denominator (with no common divisors with the numerator) is $z^{q} - c_{1} z^{q-1} - c_{2} z^{q-2} - \cdots - c_{q}$ . Between the matrices

\mathfrak {A}_{m} = \begin{pmatrix} a_{m} & a_{m+1} & \ldots & a_{m+q-1} \\ a_{m+1} & a_{m+2} & \ldots & a_{m+q} \\ \multicolumn {4}{c}{\dotfill } \\ a_{m+q-1} & a_{m+q} & \ldots & a_{m+2q-2} \end{pmatrix}, \qquad \mathfrak {C} = \begin{pmatrix} 0 & 0 & 0 & 0 & \ldots & 0 & c_{q} \\ 1 & 0 & 0 & 0 & \ldots & 0 & c_{q-1} \\ 0 & 1 & 0 & 0 & \ldots & 0 & c_{q-2} \\ 0 & 0 & 1 & 0 & \ldots & 0 & c_{q-3} \\ \multicolumn {7}{c}{\dotfill } \\ 0 & 0 & 0 & 0 & \ldots & 1 & c_{1} \end{pmatrix}

there exists the relation

\mathfrak {A}_{m} = \mathfrak {A}_{0} \mathfrak {C}^{m}, \qquad m = 0, 1, 2, \ldots .

Problem 7.33

In a system of finitely many power series there are $r$ linearly independent series if the rank of the corresponding matrix is $r$ , and any power series of the system is linearly dependent on those $r$ power series.

Remark.

Let the rank of the infinite matrix $\mathfrak {M}$ with finitely many ( $=m$ ) rows

\mathfrak {M} = \begin{pmatrix} a_{10} & a_{11} & a_{12} & \ldots & a_{1n} & \ldots \\ a_{20} & a_{21} & a_{22} & \ldots & a_{2n} & \ldots \\ \multicolumn {6}{c}{\dotfill } \\ a_{m0} & a_{m1} & a_{m2} & \ldots & a_{mn} & \ldots \end{pmatrix}

be denoted by $r_{0}$ and let the rank of the matrix that is formed from $\mathfrak {M}$ by deleting the first $n$ columns (in whose left upper corner we have the element $a_{1n}$ ) be $r_{n}$ . We have $r_{0} \geq r_{1} \geq r_{2} \geq \cdots$ , and we term $\lim_{n \rightarrow \infty } r_{n}$ the net rank of $\mathfrak {M}$ .

The matrix $\mathfrak {M}$ corresponds to the system of $m$ power series

f_{1}(z) = a_{10} + a_{11} z + a_{12} z^{2} + \cdots + a_{1n} z^{n} + \cdots

f_{2}(z) = a_{20} + a_{21} z + a_{22} z^{2} + \cdots + a_{2n} z^{n} + \cdots

f_{m}(z) = a_{m0} + a_{m1} z + a_{m2} z^{2} + \cdots + a_{mn} z^{n} + \cdots .

We say that these are linearly dependent if there exist constants $c_{1}, c_{2}, \ldots , c_{m}$ satisfying $\left|c_{1}\right| + \left|c_{2}\right| + \cdots + \left|c_{m}\right| > 0$ , such that we have

c_{1} f_{1}(z) + c_{2} f_{2}(z) + \cdots + c_{m} f_{m}(z) = 0

identically in $z$ . They are said to be quasilinearly dependent if there exist constants $c_{1}, c_{2}, \ldots , c_{m}$ satisfying $\left|c_{1}\right| + \left|c_{2}\right| + \cdots + \left|c_{m}\right| > 0$ , such that we have

c_{1} f_{1}(z) + c_{2} f_{2}(z) + \cdots + c_{m} f_{m}(z) = P(z)

identically in $z$ , where $P(z)$ is a polynomial.

The power series $f(z), zf(z), z^{2} f(z), \ldots , z^{m} f(z)$ are quasilinearly dependent for sufficiently large $m$'' and $f(z)$ represents a rational function'': these two statements are equivalent. We say that $f(z)$ represents an algebraic function if the $(m+1)^{2}$ power series $z^{\mu } [f(z)]^{\nu }$ ( $\mu , \nu = 0, 1, 2, \ldots , m$ ) are linearly dependent for sufficiently large $m$ . We say that $f(z)$ satisfies an algebraic differential equation of the $r$ th order, if the $(m+1)^{r+2}$ power series

z^{\mu } [f(z)]^{\nu } [f'(z)]^{\nu _{1}} [f''(z)]^{\nu _{2}} \ldots [f^{(r)}(z)]^{\nu _{r}} \qquad (\mu , \nu , \nu _{1}, \nu _{2}, \ldots , \nu _{r} = 0, 1, 2, \ldots , m)

are linearly dependent for sufficiently large $m$ .

Problem 7.34

In a system of finitely many power series there are $r$ quasilinearly independent series if the net rank of the corresponding matrix is $r$ , and any power series of the system is quasilinearly dependent on those $r$ power series.

Section 7.3Generation of Positive Quadratic Forms[11/100%]

Problem 7.35

Consider the two quadratic forms

\sum _{\lambda =1}^{n} \sum _{\mu =1}^{n} a_{\lambda \mu } x_{\lambda } x_{\mu } \qquad \text{and} \qquad \sum _{\lambda =1}^{n} \sum _{\mu =1}^{n} b_{\lambda \mu } x_{\lambda } x_{\mu }.

If they are positive, then the quadratic form

\sum _{\lambda =1}^{n} \sum _{\mu =1}^{n} a_{\lambda \mu } b_{\lambda \mu } x_{\lambda } x_{\mu }

is also positive. If furthermore one of the given forms is definite and if in the matrix of the other the entries on the principal diagonal are different from $0$ , then the third form is also definite.

Problem 7.36

Consider the two symmetric matrices

\begin{pmatrix} a_{11} & a_{12} & \ldots & a_{1n} \\ a_{21} & a_{22} & \ldots & a_{2n} \\ \multicolumn {4}{c}{\dotfill } \\ a_{n1} & a_{n2} & \ldots & a_{nn} \end{pmatrix}, \qquad \begin{pmatrix} e^{a_{11}} & e^{a_{12}} & \ldots & e^{a_{1n}} \\ e^{a_{21}} & e^{a_{22}} & \ldots & e^{a_{2n}} \\ \multicolumn {4}{c}{\dotfill } \\ e^{a_{n1}} & e^{a_{n2}} & \ldots & e^{a_{nn}} \end{pmatrix}.

If the quadratic form corresponding to the matrix $(a_{\lambda \mu })$ is positive, then the quadratic form corresponding to $(e^{a_{\lambda \mu }})$ is also positive. If moreover among the rows of $(a_{\lambda \mu })$ there are no two identical, then the form corresponding to $(e^{a_{\lambda \mu }})$ is in fact definite.

Remark.

See Problem 76 of Part V.

Problem 7.37

Let the power series

p_{0} + p_{1} x + p_{2} x^{2} + \cdots = F(x)

have no negative coefficients and converge for $x = a_{11}, a_{12}, \ldots , a_{nn}$ . If the quadratic form corresponding to the symmetric matrix $(a_{\lambda \mu })$ of order $n$ is positive, then the quadratic form corresponding to $(F(a_{\lambda \mu }))$ is also positive. If moreover there are among the coefficients $p_{0}, p_{2}, p_{4}, \ldots$ at least $n$ different from $0$ and if of the rows of $(a_{\lambda \mu }^{2})$ no two are identical, then the form corresponding to $(F(a_{\lambda \mu }))$ is in fact definite.

Problem 7.38

Let the real numbers $a_{0}, a_{1}, a_{2}, \ldots , a_{2n}$ have the following property: If $f(x)$ denotes an arbitrary polynomial, with real coefficients of degree at most $2n$ , that does not vanish identically and does not take a negative value for any real value of $x$ , then also

a_{0} f(x) + \frac{a_{1}}{1!} f'(x) + \frac{a_{2}}{2!} f''(x) + \cdots + \frac{a_{2n}}{(2n)!} f^{(2n)}(x) \geq 0 \qquad (\text{or} > 0)

for all real values of $x$ .

Show that for this to be true it is necessary and sufficient that the quadratic form

\sum _{\lambda =0}^{n} \sum _{\mu =0}^{n} a_{\lambda +\mu } x_{\lambda } x_{\mu }

is positive (or positive definite, respectively).

Problem 7.39

Let the numbers $a_{0}, a_{1}, \ldots , a_{n}$ , $n \geq 1$ have the following property: If $f(x)$ is an arbitrary polynomial with real coefficients of degree at most $n$ , that does not vanish identically and is non-negative for non-negative values of $x$ , then also

a_{0} f(x) + \frac{a_{1}}{1!} f'(x) + \frac{a_{2}}{2!} f''(x) + \cdots + \frac{a_{n}}{n!} f^{(n)}(x) \geq 0 \qquad (\text{or} > 0)

for all non-negative values of $x$ ( $x$ real).

Show that for this to be true it is necessary and sufficient that the two quadratic forms

\sum _{\lambda =0}^{[n/2]} \sum _{\mu =0}^{[n/2]} a_{\lambda +\mu } x_{\lambda } x_{\mu }, \qquad \sum _{\lambda =0}^{[(n-1)/2]} \sum _{\mu =0}^{[(n-1)/2]} a_{\lambda +\mu +1} x_{\lambda } x_{\mu }

are positive (or positive definite, respectively).

Problem 7.40

Let the numbers $a_{0}, a_{1}, a_{2}, \ldots , a_{n}$ , $n \geq 1$ have the following property: If $f(x)$ denotes a polynomial with real coefficients of degree $\leq n$ that does not vanish identically and is non-negative in the interval $-1 \leq x \leq 1$ , then also

a_{0} f(x) + \frac{a_{1}}{1!} f'(x) + \frac{a_{2}}{2!} f''(x) + \cdots + \frac{a_{n}}{n!} f^{(n)}(x) \geq 0

for $-1 \leq x \leq 1$ . Show that the only set of numbers $a_{0}, a_{1}, a_{2}, \ldots , a_{n}$ of this kind is determined by the conditions

a_{0} \geq 0, \qquad a_{1} = a_{2} = \cdots = a_{n} = 0.

Problem 7.41

Let the two quadratic forms

\sum _{\lambda =0}^{n} \sum _{\mu =0}^{n} a_{\lambda +\mu } x_{\lambda } x_{\mu }, \qquad \sum _{\lambda =0}^{n} \sum _{\mu =0}^{n} b_{\lambda +\mu } x_{\lambda } x_{\mu }

be positive. If we set

c_{\nu } = a_{0} b_{\nu } + \binom {\nu }{1} a_{1} b_{\nu -1} + \binom {\nu }{2} a_{2} b_{\nu -2} + \cdots + a_{\nu } b_{0},

then the quadratic form

\sum _{\lambda =0}^{n} \sum _{\mu =0}^{n} c_{\lambda +\mu } x_{\lambda } x_{\mu }

is also positive, and it is in fact definite if at least one of the two preceding forms is definite and the other not identically vanishing.

Problem 7.42

Let

c_{\nu } = a_{0} b_{\nu } + \binom {\nu }{1} a_{1} b_{\nu -1} + \binom {\nu }{2} a_{2} b_{\nu -2} + \cdots + a_{\nu } b_{0}.

From the positiveness of the four quadratic forms

\sum _{\lambda =0}^{m} \sum _{\mu =0}^{m} a_{\lambda +\mu } x_{\lambda } x_{\mu }, \qquad \sum _{\lambda =0}^{m-1} \sum _{\mu =0}^{m-1} a_{\lambda +\mu +1} x_{\lambda } x_{\mu },

\sum _{\lambda =0}^{m} \sum _{\mu =0}^{m} b_{\lambda +\mu } x_{\lambda } x_{\mu }, \qquad \sum _{\lambda =0}^{m-1} \sum _{\mu =0}^{m-1} b_{\lambda +\mu +1} x_{\lambda } x_{\mu }

follows the positiveness of the following two forms:

\sum _{\lambda =0}^{m} \sum _{\mu =0}^{m} c_{\lambda +\mu } x_{\lambda } x_{\mu }, \qquad \sum _{\lambda =0}^{m-1} \sum _{\mu =0}^{m-1} c_{\lambda +\mu +1} x_{\lambda } x_{\mu }.

Furthermore, from the positiveness of the four quadratic forms

\sum _{\lambda =0}^{m} \sum _{\mu =0}^{m} a_{\lambda +\mu } x_{\lambda } x_{\mu }, \qquad \sum _{\lambda =0}^{m} \sum _{\mu =0}^{m} a_{\lambda +\mu +1} x_{\lambda } x_{\mu },

\sum _{\lambda =0}^{m} \sum _{\mu =0}^{m} b_{\lambda +\mu } x_{\lambda } x_{\mu }, \qquad \sum _{\lambda =0}^{m} \sum _{\mu =0}^{m} b_{\lambda +\mu +1} x_{\lambda } x_{\mu }

follows the positiveness of the following two forms:

\sum _{\lambda =0}^{m} \sum _{\mu =0}^{m} c_{\lambda +\mu } x_{\lambda } x_{\mu }, \qquad \sum _{\lambda =0}^{m} \sum _{\mu =0}^{m} c_{\lambda +\mu +1} x_{\lambda } x_{\mu }.

Problem 7.43

Let $\vartheta$ be real and let the complex numbers

c_{-n}, c_{-n+1}, \ldots , c_{-1}, c_{0}, c_{1}, \ldots , c_{n-1}, c_{n}, \qquad c_{-\nu } = \bar{c}_{\nu }, \quad \nu = 0, 1, 2, \ldots , n

have the following property: If

g(\vartheta ) = \alpha _{0} + 2(\alpha _{1} \cos \vartheta + \beta _{1} \sin \vartheta + \alpha _{2} \cos 2\vartheta + \beta _{2} \sin 2\vartheta + \cdots + \alpha _{n} \cos n\vartheta + \beta _{n} \sin n\vartheta )

= \sum _{\nu =-n}^{n} \gamma _{\nu } e^{-i\nu \vartheta }, \qquad \gamma _{\nu } = \bar{\gamma }_{-\nu } = \alpha _{\nu } + i\beta _{\nu }, \quad \nu = 0, 1, 2, \ldots , n; \quad \beta _{0} = 0,

denotes an arbitrary trigonometric polynomial of degree at most $n$ , that does not vanish identically and does not have a negative value for any value of $\vartheta$ , let

\sum _{\nu =-n}^{n} c_{\nu } \gamma _{\nu } e^{-i\nu \vartheta } \geq 0 \qquad (\text{or} > 0).

Show that for this to be true, it is necessary and sufficient that the Hermitian form

\sum _{\lambda =0}^{n} \sum _{\mu =0}^{n} c_{\mu -\lambda } x_{\lambda } \bar{x}_{\mu }

is positive (or positive definite, respectively).

Problem 7.43.1

Let $G(z)$ be an entire function that takes real values for real $z$ , with $G(0) = 1$ , and

-\frac{G'(z)}{G(z)} = s_{1} + s_{2} z + s_{3} z^{2} + \cdots + s_{m} z^{m-1} + \cdots .

If $G(z)$ has infinitely many zeros, all real, and is of the form given in Problem 165 of Part V, then, for $n = 1, 2, 3, \ldots$ ,

\begin{vmatrix} s_{2} & s_{3} & s_{4} & \ldots & s_{n+1} \\ s_{3} & s_{4} & s_{5} & \ldots & s_{n+2} \\ s_{4} & s_{5} & s_{6} & \ldots & s_{n+3} \\ \multicolumn {5}{c}{\dotfill } \\ s_{n+1} & s_{n+2} & s_{n+3} & \ldots & s_{2n} \end{vmatrix} > 0.

(If $G(z)$ of the form in Problem 165 of Part V has only a finite number of zeros, then only a finite number of these recurrent determinants can be $\neq 0$ [Problem 27].)

Problem 7.43.1

Let $G(z)$ be an entire function that takes real values for real $z$ , with $G(0) = 1$ , and

-\frac{G'(z)}{G(z)} = s_{1} + s_{2} z + s_{3} z^{2} + \cdots + s_{m} z^{m-1} + \cdots .

If $G(z)$ has infinitely many zeros, all real, and is of the form given in Problem 165 of Part V, then, for $n = 1, 2, 3, \ldots$ ,

\begin{vmatrix} s_{2} & s_{3} & s_{4} & \ldots & s_{n+1} \\ s_{3} & s_{4} & s_{5} & \ldots & s_{n+2} \\ s_{4} & s_{5} & s_{6} & \ldots & s_{n+3} \\ \multicolumn {5}{c}{\dotfill } \\ s_{n+1} & s_{n+2} & s_{n+3} & \ldots & s_{2n} \end{vmatrix} > 0.

(If $G(z)$ of the form in Problem 165 of Part V has only a finite number of zeros, then only a finite number of these recurrent determinants can be $\neq 0$ [Problem 27].)

Problem 7.43.2

Let $G(z)$ be an entire function of the form given in Problem 165 of Part V, but not of the form $e^{\beta z}$ , and

\frac{1}{G(z)} = c_{0} + \frac{c_{1}}{1!} z + \frac{c_{2}}{2!} z^{2} + \frac{c_{3}}{3!} z^{3} + \cdots .

Then, for $n = 0, 1, 2, \ldots$ ,

\begin{vmatrix} c_{0} & c_{1} & c_{2} & \ldots & c_{n} \\ c_{1} & c_{2} & c_{3} & \ldots & c_{n+1} \\ c_{2} & c_{3} & c_{4} & \ldots & c_{n+2} \\ \multicolumn {5}{c}{\dotfill } \\ c_{n} & c_{n+1} & c_{n+2} & \ldots & c_{2n} \end{vmatrix} > 0.

Problem 7.43.2

Let $G(z)$ be an entire function of the form given in Problem 165 of Part V, but not of the form $e^{\beta z}$ , and

\frac{1}{G(z)} = c_{0} + \frac{c_{1}}{1!} z + \frac{c_{2}}{2!} z^{2} + \frac{c_{3}}{3!} z^{3} + \cdots .

Then, for $n = 0, 1, 2, \ldots$ ,

\begin{vmatrix} c_{0} & c_{1} & c_{2} & \ldots & c_{n} \\ c_{1} & c_{2} & c_{3} & \ldots & c_{n+1} \\ c_{2} & c_{3} & c_{4} & \ldots & c_{n+2} \\ \multicolumn {5}{c}{\dotfill } \\ c_{n} & c_{n+1} & c_{n+2} & \ldots & c_{2n} \end{vmatrix} > 0.

Section 7.4Miscellaneous Problems[17/100%]

Problem 7.44

Let the $n^{2}$ elements $a_{\lambda \mu }$ of a determinant of $n$ th order be independent variables. Show that, of the $n!$ terms $\pm a_{1k_{1}} a_{2k_{2}} \ldots a_{nk_{n}}$ in the expansion of the determinant, only $N = n^{2} - 2n + 2$ are independent, and determine $N$ terms by means of which all the remaining terms may be expressed rationally.

Problem 7.45

In the expansion of a symmetric $n$ th order determinant (with arbitrary elements) let $s_{n}'$ be the number of distinct positive terms and $s_{n}''$ the number of distinct negative terms. For $s_{n} = s_{n}' + s_{n}''$ we have already in the works of Cayley the recurrence relation

s_{n+1} = (n+1) s_{n} - \binom {n}{2} s_{n-2}.

Show that for $d_{n} = s_{n}' - s_{n}''$ we have the recurrence relation

d_{n+1} = -(n-1) d_{n} - \binom {n}{2} d_{n-2},

and that

\lim _{n \rightarrow \infty } \frac{n^{1/2} s_{n}}{n!} = \frac{e^{3/4}}{\sqrt{\pi }}, \qquad \lim _{n \rightarrow \infty } \frac{(-1)^{n-1} n^{3/2} d_{n}}{n!} = \frac{e^{-3/4}}{2\sqrt{\pi }}.

Problem 7.46

In the expansion of a symmetric $n$ th order determinant $\left|a_{\lambda \mu }\right|$ , in which the $n$ elements $a_{\lambda \lambda }$ in the leading diagonal are equal to zero, let $\sigma_{n}'$ be the number of distinct positive terms and $\sigma_{n}''$ the number of distinct negative terms. Setting

\sigma _{n} = \sigma _{n}' + \sigma _{n}'', \qquad \delta _{n} = \sigma _{n}' - \sigma _{n}'',

we have

\sigma _{n+1} = n\sigma _{n} + n\sigma _{n-1} - \binom {n}{2} \sigma _{n-2}, \qquad \delta _{n+1} = -n\delta _{n} - n\delta _{n-1} - \binom {n}{2} \delta _{n-2}

and

\lim _{n \rightarrow \infty } \frac{n^{1/2} \sigma _{n}}{n!} = \frac{e^{-1/4}}{\sqrt{\pi }}, \qquad \lim _{n \rightarrow \infty } \frac{(-1)^{n-1} n^{3/2} \delta _{n}}{n!} = \frac{e^{1/4}}{2\sqrt{\pi }}.

Problem 7.46.1

Let the capital letters $A, B, \ldots , I, \ldots , Z$ denote $n \times n$ matrices. In particular, let $I$ denote the identity matrix with entries $1$ along the main diagonal and $0$ elsewhere, and $Z$ denote the matrix in which all entries are $1$ . Show that

Z \cdot Z = n \cdot Z.

Problem 7.46.1

Z \cdot Z = n \cdot Z.

Problem 7.46.2

(continued). If all the entries of $A$ are positive numbers,

A \cdot B = I,

and $n > 1$ , then there are among the entries of $B$ at least $n$ positive numbers and at least $n$ negative numbers.

Prove this, and show by examples that the two bounds for the number of positive entries, $n$ and $n^{2} - n$ , can be attained.

Problem 7.46.2

(continued). If all the entries of $A$ are positive numbers,

A \cdot B = I,

and $n > 1$ , then there are among the entries of $B$ at least $n$ positive numbers and at least $n$ negative numbers.

Prove this, and show by examples that the two bounds for the number of positive entries, $n$ and $n^{2} - n$ , can be attained.

Problem 7.47

Denote the difference product $\prod_{j>k}^{1,2,\ldots ,n} (x_{j} - x_{k})$ by $\Delta$ . It is well known that an expression of the form

\Phi = \frac{1}{\Delta } \sum \pm \varphi (x_{\lambda _{1}}, x_{\lambda _{2}}, \ldots , x_{\lambda _{n}})

is a symmetric function of $x_{1}, x_{2}, \ldots , x_{n}$ . Here the sum is to be taken over all the $n!$ permutations of $1, 2, \ldots , n$ and for the even permutations the positive sign is to be chosen and for the odd permutations the negative sign. Prove that, if

\varphi = \prod _{\nu =1}^{n} \frac{1}{1 - x_{1} x_{2} \ldots x_{\nu }},

then

\Phi = \frac{1}{\prod _{\nu =1}^{n} (1-x_{\nu }) \prod _{j>k}^{1,2,\ldots ,n} (1 - x_{j} x_{k})}.

Problem 7.48

Let the characteristic equation of the matrix $\left\| a_{\lambda \mu }\right\|$ , $\lambda , \mu = 1, 2, \ldots , n$ ,

\chi (z) = \begin{vmatrix} a_{11} - z & a_{12} & \ldots & a_{1n} \\ a_{21} & a_{22} - z & \ldots & a_{2n} \\ \multicolumn {4}{c}{\dotfill } \\ a_{n1} & a_{n2} & \ldots & a_{nn} - z \end{vmatrix} = 0

have the roots $\alpha_{1}, \alpha_{2}, \ldots , \alpha_{n}$ , that need not necessarily be all distinct. Denote the minors of order $n-1$ of $\chi (z)$ by $\chi_{\lambda \mu }(z)$ . Prove that the characteristic equation of the matrix $\chi_{\lambda \mu }(z)$ has the roots

\zeta _{\rho } = \frac{\chi (z)}{\alpha _{\rho } - z}, \qquad \rho = 1, 2, \ldots , n.

Problem 7.49

The linear transformations

S_{\alpha }: \qquad y_{p} = \frac{\sin \alpha \pi }{n+1} \sum _{q=0}^{n} \frac{x_{q}}{\sin \dfrac {\pi }{n+1} (p-q+\alpha )}, \qquad p = 0, 1, \ldots , n,

$\alpha$ arbitrary, form a group in the following sense: We have

S_{\alpha } S_{\beta } = S_{\alpha +\beta }.

Problem 7.50

Let $g(\vartheta )$ be a trigonometric polynomial of order at most $n$ , with purely real coefficients. Determine the conditions that $g(\vartheta )$ must satisfy, in order that the linear transformations

S_{\alpha }: \qquad y_{p} = \sum _{q=0}^{n} x_{q} g\left( \frac{\pi }{n+1} (p-q+\alpha ) \right), \qquad p = 0, 1, \ldots , n,

$\alpha$ arbitrary, form a group in the following sense: We have

S_{\alpha } S_{\beta } = S_{\alpha +\beta }.

Problem 7.51

(continued). The determinant of the linear transformation $S_{\alpha }$ vanishes for all values of $\alpha$ . The single exceptional case, when it is identically $=1$ , is

g(\vartheta ) = \frac{1}{n+1} \frac{\sin (n+1)\vartheta }{\sin \vartheta }.

Problem 7.52

(Continuation of Problem 49.) The transformation $S_{\alpha }$ is orthogonal, i.e. we have identically in $x_{0}, x_{1}, x_{2}, \ldots , x_{n}$ that

y_{0}^{2} + y_{1}^{2} + y_{2}^{2} + \cdots + y_{n}^{2} = x_{0}^{2} + x_{1}^{2} + x_{2}^{2} + \cdots + x_{n}^{2}.

We say that the linear transformation

\begin{aligned} y_{1} & = a_{11} x_{1} + a_{12} x_{2} + \cdots + a_{1n} x_{n} + \cdots \\ y_{2} & = a_{21} x_{1} + a_{22} x_{2} + \cdots + a_{2n} x_{n} + \cdots \\ & \phantom{{}={}} \dotfill \\ y_{m} & = a_{m1} x_{1} + a_{m2} x_{2} + \cdots + a_{mn} x_{n} + \cdots \\ & \phantom{{}={}} \dotfill \end{aligned}

or that the corresponding matrix $\left\| a_{mn}\right\|$ is orthogonal, if the relations

a_{m1}^{2} + a_{m2}^{2} + a_{m3}^{2} + \cdots + a_{mn}^{2} + \cdots = 1, \qquad m = 1, 2, 3, \ldots ,

a_{\lambda 1} a_{\mu 1} + a_{\lambda 2} a_{\mu 2} + a_{\lambda 3} a_{\mu 3} + \cdots + a_{\lambda n} a_{\mu n} + \cdots = 0, \qquad \lambda \leq \mu , \quad \lambda , \mu = 1, 2, 3, \ldots

and also the further relations

a_{1n}^{2} + a_{2n}^{2} + a_{3n}^{2} + \cdots + a_{mn}^{2} + \cdots = 1, \qquad n = 1, 2, 3, \ldots ,

a_{1\lambda } a_{1\mu } + a_{2\lambda } a_{2\mu } + a_{3\lambda } a_{3\mu } + \cdots + a_{m\lambda } a_{m\mu } + \cdots = 0, \qquad \lambda \leq \mu , \quad \lambda , \mu = 1, 2, 3, \ldots

are satisfied. We define similarly the orthogonality of a matrix that goes to infinity in four directions:

\begin{pmatrix} \multicolumn {7}{c}{\dotfill } \\ \ldots & a_{-m,-n} & \ldots & a_{-m,-1} & a_{-m,0} & a_{-m,1} & \ldots \\ \multicolumn {7}{c}{\dotfill } \\ \ldots & a_{m,-n} & \ldots & a_{m,-1} & a_{m,0} & a_{m,1} & \ldots \\ \multicolumn {7}{c}{\dotfill } \end{pmatrix}.

Problem 7.53

The Fibonacci numbers $0, 1, 1, 2, 3, 5, 8, 13, 21, 34, \ldots$ are defined as follows: Set $u_{0} = 0$ , $u_{1} = 1$ , $u_{n} + u_{n+1} = u_{n+2}$ for $n = 0, 1, 2, \ldots$ . The linear transformation

y_{n} = \frac{u_{1} x_{1} + u_{2} x_{2} + \cdots + u_{n} x_{n} - u_{n} x_{n+1}}{\sqrt{u_{n} u_{n+2}}}, \qquad n = 1, 2, 3, \ldots

is orthogonal.

Problem 7.54

Let $\alpha$ be real and not integral. The linear transformation

y_{m} = \frac{\sin \alpha \pi }{\pi } \sum _{n=-\infty }^{\infty } \frac{x_{n}}{m+n-\alpha }, \qquad m = \ldots , -2, -1, 0, 1, 2, \ldots

is orthogonal.

Problem 7.54.1

In problems 54.1--54.4 we consider infinite triangular matrices

\begin{pmatrix} a_{11} & & & \\ a_{21} & a_{22} & & \\ a_{31} & a_{32} & a_{33} & \\ \multicolumn {4}{c}{\dotfill } \end{pmatrix},

where it is understood that all elements above the principal diagonal are $0$ . An equation between such matrices is regarded as valid if, and only if, it is valid between the corresponding submatrices with $n^{2}$ elements obtained by deleting all rows and columns except the first $n$ , for $n = 1, 2, 3, \ldots$ .

Prove the relation, involving binomial coefficients,

\begin{pmatrix} 1 & & & \\ 1 & 1 & & \\ 1 & 2 & 1 & \\ 1 & 3 & 3 & 1 \\ \multicolumn {4}{c}{\dotfill } \end{pmatrix} \begin{pmatrix} 1 & & & \\ -1 & 1 & & \\ 1 & -2 & 1 & \\ -1 & 3 & -3 & 1 \\ \multicolumn {4}{c}{\dotfill } \end{pmatrix} = \begin{pmatrix} 1 & & & \\ 0 & 1 & & \\ 0 & 0 & 1 & \\ 0 & 0 & 0 & 1 \\ \multicolumn {4}{c}{\dotfill } \end{pmatrix}.

Problem 7.54.1

In problems 54.1--54.4 we consider infinite triangular matrices

\begin{pmatrix} a_{11} & & & \\ a_{21} & a_{22} & & \\ a_{31} & a_{32} & a_{33} & \\ \multicolumn {4}{c}{\dotfill } \end{pmatrix},

Prove the relation, involving binomial coefficients,

\begin{pmatrix} 1 & & & \\ 1 & 1 & & \\ 1 & 2 & 1 & \\ 1 & 3 & 3 & 1 \\ \multicolumn {4}{c}{\dotfill } \end{pmatrix} \begin{pmatrix} 1 & & & \\ -1 & 1 & & \\ 1 & -2 & 1 & \\ -1 & 3 & -3 & 1 \\ \multicolumn {4}{c}{\dotfill } \end{pmatrix} = \begin{pmatrix} 1 & & & \\ 0 & 1 & & \\ 0 & 0 & 1 & \\ 0 & 0 & 0 & 1 \\ \multicolumn {4}{c}{\dotfill } \end{pmatrix}.

Problem 7.54.2

Prove the relation, involving Stirling numbers $s_{k}^{n}$ and $S_{k}^{n}$ of the first and second kind respectively (see the introductions to Problems 197 and 186 of Part I),

\begin{pmatrix} 1 & & & \\ 1 & 1 & & \\ 1 & 3 & 1 & \\ 1 & 7 & 6 & 1 \\ \multicolumn {4}{c}{\dotfill } \end{pmatrix} \begin{pmatrix} 1 & & & \\ -1 & 1 & & \\ 2 & -3 & 1 & \\ -6 & 11 & -6 & 1 \\ \multicolumn {4}{c}{\dotfill } \end{pmatrix} = \begin{pmatrix} 1 & & & \\ 0 & 1 & & \\ 0 & 0 & 1 & \\ 0 & 0 & 0 & 1 \\ \multicolumn {4}{c}{\dotfill } \end{pmatrix}.

In other words, the matrices $(S_{k}^{n})$ and $((-1)^{n-k} s_{k}^{n})$ are inverse to each other.

Problem 7.54.2

Prove the relation, involving Stirling numbers $s_{k}^{n}$ and $S_{k}^{n}$ of the first and second kind respectively (see the introductions to Problems 197 and 186 of Part I),

\begin{pmatrix} 1 & & & \\ 1 & 1 & & \\ 1 & 3 & 1 & \\ 1 & 7 & 6 & 1 \\ \multicolumn {4}{c}{\dotfill } \end{pmatrix} \begin{pmatrix} 1 & & & \\ -1 & 1 & & \\ 2 & -3 & 1 & \\ -6 & 11 & -6 & 1 \\ \multicolumn {4}{c}{\dotfill } \end{pmatrix} = \begin{pmatrix} 1 & & & \\ 0 & 1 & & \\ 0 & 0 & 1 & \\ 0 & 0 & 0 & 1 \\ \multicolumn {4}{c}{\dotfill } \end{pmatrix}.

In other words, the matrices $(S_{k}^{n})$ and $((-1)^{n-k} s_{k}^{n})$ are inverse to each other.

Problem 7.54.3

Prove

\begin{pmatrix} 1 & & & & \\ 1 & -1 & & & \\ 1 & -2 & 1 & & \\ 1 & -3 & 3 & -1 & \\ 1 & -4 & 6 & -4 & 1 \\ \multicolumn {5}{c}{\dotfill } \end{pmatrix}^{2} = \begin{pmatrix} 1 & & & & \\ 0 & 1 & & & \\ 0 & 0 & 1 & & \\ 0 & 0 & 0 & 1 & \\ 0 & 0 & 0 & 0 & 1 \\ \multicolumn {5}{c}{\dotfill } \end{pmatrix}.

Remark.

Cf. Problem 34.1 of Part I and Problem 54.1.

Problem 7.54.3

Prove

\begin{pmatrix} 1 & & & & \\ 1 & -1 & & & \\ 1 & -2 & 1 & & \\ 1 & -3 & 3 & -1 & \\ 1 & -4 & 6 & -4 & 1 \\ \multicolumn {5}{c}{\dotfill } \end{pmatrix}^{2} = \begin{pmatrix} 1 & & & & \\ 0 & 1 & & & \\ 0 & 0 & 1 & & \\ 0 & 0 & 0 & 1 & \\ 0 & 0 & 0 & 0 & 1 \\ \multicolumn {5}{c}{\dotfill } \end{pmatrix}.

Remark.

Cf. Problem 34.1 of Part I and Problem 54.1.

Problem 7.54.4

Generalizing Problem 54.3, find all infinite sequences of numbers $a_{0}, a_{1}, a_{2}, \ldots$ such that the system of relations

y_{n} = a_{n} x_{0} - \binom {n}{1} a_{n-1} x_{1} + \binom {n}{2} a_{n-2} x_{2} - \cdots + (-1)^{n} a_{0} x_{n}, \qquad n = 0, 1, 2, 3, \ldots ,

should imply the same system with the roles of $x$ and $y$ reversed, that is,

x_{n} = a_{n} y_{0} - \binom {n}{1} a_{n-1} y_{1} + \binom {n}{2} a_{n-2} y_{2} - \cdots + (-1)^{n} a_{0} y_{n}

for $n = 0, 1, 2, 3, \ldots$ .

Remark.

See Problem 34.1 of Part I.

Problem 7.54.4

Generalizing Problem 54.3, find all infinite sequences of numbers $a_{0}, a_{1}, a_{2}, \ldots$ such that the system of relations

y_{n} = a_{n} x_{0} - \binom {n}{1} a_{n-1} x_{1} + \binom {n}{2} a_{n-2} x_{2} - \cdots + (-1)^{n} a_{0} x_{n}, \qquad n = 0, 1, 2, 3, \ldots ,

should imply the same system with the roles of $x$ and $y$ reversed, that is,

x_{n} = a_{n} y_{0} - \binom {n}{1} a_{n-1} y_{1} + \binom {n}{2} a_{n-2} y_{2} - \cdots + (-1)^{n} a_{0} y_{n}

for $n = 0, 1, 2, 3, \ldots$ .

Remark.

See Problem 34.1 of Part I.

Section 7.5Determinants of Systems of Functions[18/100%]

Problem 7.55

We term the determinant

\begin{vmatrix} f_{1}(x) & f_{1}'(x) & f_{1}''(x) & \ldots & f_{1}^{(n-1)}(x) \\ f_{2}(x) & f_{2}'(x) & f_{2}''(x) & \ldots & f_{2}^{(n-1)}(x) \\ f_{3}(x) & f_{3}'(x) & f_{3}''(x) & \ldots & f_{3}^{(n-1)}(x) \\ \multicolumn {5}{c}{\dotfill } \\ f_{n}(x) & f_{n}'(x) & f_{n}''(x) & \ldots & f_{n}^{(n-1)}(x) \end{vmatrix} = W[f_{1}(x), f_{2}(x), \ldots , f_{n}(x)]

the Wronskian determinant of the system of functions $f_{1}(x), f_{2}(x), \ldots , f_{n}(x)$ .

If $c_{\lambda \mu }$ are constants, then we have

W(c_{11} f_{1} + c_{12} f_{2} + \cdots + c_{1n} f_{n}, \; c_{21} f_{1} + c_{22} f_{2} + \cdots + c_{2n} f_{n}, \; \ldots , \; c_{n1} f_{1} + c_{n2} f_{2} + \cdots + c_{nn} f_{n}) = \left|c_{\lambda \mu }\right|_{1}^{n} \cdot W(f_{1}, f_{2}, \ldots , f_{n}).

Problem 7.56

W[f_{1}(\varphi (x)), f_{2}(\varphi (x)), \ldots , f_{n}(\varphi (x))] = \varphi '(x)^{\frac{n(n-1)}{2}} W[f_{1}(y), f_{2}(y), \ldots , f_{n}(y)],

setting $y = \varphi (x)$ on the right-hand side.

Problem 7.57

W(\varphi f_{1}, \varphi f_{2}, \ldots , \varphi f_{n}) = \varphi ^{n} W(f_{1}, f_{2}, \ldots , f_{n}).

Problem 7.58

\frac{1}{f_{1}^{n}} W(f_{1}, f_{2}, \ldots , f_{n}) = W\left[ \left( \frac{f_{2}}{f_{1}} \right)', \left( \frac{f_{3}}{f_{1}} \right)', \ldots , \left( \frac{f_{n}}{f_{1}} \right)' \right].

Problem 7.59

\frac{d}{dx} \frac{W(f_{1}, \ldots , f_{n-2}, f_{n})}{W(f_{1}, \ldots , f_{n-2}, f_{n-1})} = \frac{W(f_{1}, \ldots , f_{n-2}) W(f_{1}, \ldots , f_{n-2}, f_{n-1}, f_{n})}{[W(f_{1}, \ldots , f_{n-2}, f_{n-1})]^{2}}.

Problem 7.60

If $W(f_{1}, f_{2}, \ldots , f_{n-1}, f_{n})$ vanishes at every point and $W(f_{1}, f_{2}, \ldots , f_{n-1})$ at no point of the interval $a, b$ , then there exist $n-1$ constants $c_{1}, c_{2}, \ldots , c_{n-1}$ such that in the whole interval $a, b$

f_{n}(x) = c_{1} f_{1}(x) + c_{2} f_{2}(x) + \cdots + c_{n-1} f_{n-1}(x).

Problem 7.61

If $f_{1}(x), f_{2}(x), \ldots , f_{n}(x)$ are $n$ linearly independent solutions of the homogeneous linear $n$ th order differential equation

y^{(n)} + \varphi _{1}(x) y^{(n-1)} + \varphi _{2}(x) y^{(n-2)} + \cdots + \varphi _{n}(x) y = 0,

then we have for an arbitrary function $y$

y^{(n)} + \varphi _{1}(x) y^{(n-1)} + \varphi _{2}(x) y^{(n-2)} + \cdots + \varphi _{n}(x) y = \frac{W(f_{1}, f_{2}, \ldots , f_{n}, y)}{W(f_{1}, f_{2}, \ldots , f_{n})}.

Problem 7.62

(continued). Set

1 = W_{0}, \qquad f_{1} = W_{1}, \qquad W(f_{1}, f_{2}) = W_{2}, \qquad \ldots , \qquad W(f_{1}, f_{2}, \ldots , f_{n}) = W_{n}.

Then we have for an arbitrary function $y$

y^{(n)} + \varphi _{1}(x) y^{(n-1)} + \varphi _{2}(x) y^{(n-2)} + \cdots + \varphi _{n}(x) y

\equiv \frac{W_{n}}{W_{n-1}} \cdot \frac{d}{dx} \frac{W_{n-1}^{2}}{W_{n-2} W_{n}} \cdots \frac{d}{dx} \frac{W_{2}^{2}}{W_{1} W_{3}} \cdot \frac{d}{dx} \frac{W_{1}^{2}}{W_{0} W_{2}} \cdot \frac{d}{dx} \frac{y}{W_{1}}.

This representation of the linear differential expression, in terms of $n$ independent solutions of the homogeneous differential equation, is analogous to the factorization of a polynomial of degree $n$ in terms of its $n$ zeros.

Problem 7.63

Let $f_{1}(x), f_{2}(x), \ldots , f_{n}(x)$ be continuous real functions, defined in the interval $a, b$ . The determinant

\left|\int _{a}^{b} f_{\lambda }(x) f_{\mu }(x) \, dx\right|_{\lambda , \mu = 1, 2, \ldots , n}

is never negative. It vanishes if and only if between the functions $f_{1}(x), f_{2}(x), \ldots , f_{n}(x)$ there exists a homogeneous linear relation with constant coefficients, not all $0$ .

Problem 7.64

(continued). The determinant

\begin{vmatrix} \displaystyle \int _{a}^{b} (f_{2}^{2}+f_{3}^{2}+\cdots +f_{n}^{2}) \, dx & -\displaystyle \int _{a}^{b} f_{1} f_{2} \, dx & \ldots & -\displaystyle \int _{a}^{b} f_{1} f_{n} \, dx \\ ]2ex] -\displaystyle \int _{a}^{b} f_{2} f_{1} \, dx & \displaystyle \int _{a}^{b} (f_{1}^{2}+f_{3}^{2}+\cdots +f_{n}^{2}) \, dx & \ldots & -\displaystyle \int _{a}^{b} f_{2} f_{n} \, dx \\[2ex] \multicolumn {4}{c}{\dotfill } \\ ]1ex] -\displaystyle \int _{a}^{b} f_{n} f_{1} \, dx & -\displaystyle \int _{a}^{b} f_{n} f_{2} \, dx & \ldots & \displaystyle \int _{a}^{b} (f_{1}^{2}+f_{2}^{2}+\cdots +f_{n-1}^{2}) \, dx \end{vmatrix} \[ is never negative. It vanishes if and only if the functions $f_{1}(x), f_{2}(x), \ldots , f_{n}(x)$ differ from one another only by constant factors, or, in other words, if they are all multiples of one of them. \]

Problem 7.65

(continued). The determinant

\left|e^{\int _{a}^{b} f_{\lambda }(x) f_{\mu }(x) \, dx}\right|_{\lambda , \mu = 1, 2, \ldots , n}

is never negative. It vanishes if and only if among the functions $f_{1}(x), f_{2}(x), \ldots , f_{n}(x)$ there are two identically equal.

Problem 7.66

A theorem mentioned in passing by Gauss in his Theoria combinationis observationum [Werke, Vol. 4, p. 12, cf. the last two lines of Art. 11] may be generalized as follows:

Let $f(x)$ be defined for $x > 0$ , non-increasing there, $f(x) \geq 0$ , but not identically $=0$ . Moreover let the real numbers $a_{1}, a_{2}, \ldots , a_{n}$ be distinct. Assuming the existence of all the integrals that occur, the determinant

\left|(a_{\lambda }+a_{\mu }+1) \int _{0}^{\infty } x^{a_{\lambda }+a_{\mu }} f(x) \, dx\right|_{\lambda , \mu = 1, 2, \ldots , n}

is never negative. It is $=0$ if and only if $f(x)$ is a piecewise constant function with not more than $n-1$ jump discontinuities.

Problem 7.67

Let the function $f(\vartheta )$ be properly integrable in the interval $0 \leq \vartheta < 2\pi$ . Let its Fourier series (Section 4 of Part VI) be

f(\vartheta ) \sim a_{0} + 2 \sum _{n=1}^{\infty } (a_{n} \cos n\vartheta + b_{n} \sin n\vartheta ).

The Hermite forms

T_{n}(f) = \sum _{\lambda =0}^{n} \sum _{\mu =0}^{n} c_{\mu -\lambda } x_{\lambda } \bar{x}_{\mu }

formed from the constants $c_{0} = a_{0}$ , $c_{n} = a_{n} + ib_{n}$ , $c_{-n} = \bar{c}_{n}$ , $n = 1, 2, 3, \ldots$ , are called the Toeplitz forms associated with $f(\vartheta )$ . (Cf. Problem 43).

Form the Toeplitz forms that belong to the following functions:

f(\vartheta ) = c = \text{const.}, \qquad f(\vartheta ) = a_{0} + 2(a_{1} \cos \vartheta + b_{1} \sin \vartheta ),

f(\vartheta ) = \frac{1-r^{2}}{1-2r\cos \vartheta +r^{2}}, \qquad 0 < r < 1.

Problem 7.68

If the function $f(\vartheta )$ is positive in the interval $0 \leq \vartheta < 2\pi$ , then all its associated Toeplitz forms are positive definite. More precisely: If $f(\vartheta )$ lies between the bounds $m \leq f(\vartheta ) \leq M$ in the interval $0 \leq \vartheta < 2\pi$ , then we have

m \leq T_{n}(f) \leq M,

provided that the variables $x_{0}, x_{1}, \ldots , x_{n}$ satisfy the condition $T_{n}(1) = 1$ . The equality sign can hold here only if $f(\vartheta ) = \text{const.}$ at every point of continuity of $f(\vartheta )$ .

Problem 7.69

Calculate the determinant $D_{n}(f)$ of the form $T_{n}(f)$ for the functions in Problem 67 and show that, if $f(\vartheta )$ is everywhere positive,

\lim _{n \rightarrow \infty } \sqrt[n+1]{D_{n}(f)} = e^{\frac{1}{2\pi } \int _{0}^{2\pi } \log f(\vartheta ) \, d\vartheta } = \mathfrak {G}(f).

Problem 7.70

Let $f(\vartheta )$ be a trigonometric polynomial of first order and let $h$ be a parameter. The determinant $D_{n}(f-h)$ of the Toeplitz form $T_{n}(f-h)$ associated with $f(\vartheta )-h$ is a polynomial of degree $n+1$ in $h$ with only real zeros. Calculate these zeros.

Problem 7.71

(continued). The zeros $h_{0n}, h_{1n}, \ldots , h_{nn}$ of $D_{n}(f-h)$ all lie between the minimum and the maximum of $f(\vartheta )$ , that is $m \leq h_{\nu n} \leq M$ , if $m \leq f(\vartheta ) \leq M$ in the interval $0, 2\pi$ . Moreover if $F(h)$ denotes an arbitrary properly integrable function in the interval $m \leq h \leq M$ , then we have

\lim _{n \rightarrow \infty } \frac{F(h_{0n})+F(h_{1n})+\cdots +F(h_{nn})}{n+1} = \frac{1}{2\pi } \int _{0}^{2\pi } F[f(\vartheta )] \, d\vartheta .

Problem 7.72

Let the Hermitian form

\sum _{\lambda =0}^{n} \sum _{\mu =0}^{n} c_{\lambda -\mu } x_{\lambda } \bar{x}_{\mu }, \qquad c_{-\nu } = \bar{c}_{\nu },

be positive definite. Then all the zeros of the polynomial

\begin{vmatrix} c_{0} & c_{1} & c_{2} & \ldots & c_{n} \\ c_{-1} & c_{0} & c_{1} & \ldots & c_{n-1} \\ c_{-2} & c_{-1} & c_{0} & \ldots & c_{n-2} \\ \multicolumn {5}{c}{\dotfill } \\ c_{-n+1} & c_{-n+2} & c_{-n+3} & \ldots & c_{1} \\ 1 & z & z^{2} & \ldots & z^{n} \end{vmatrix}

lie in the interior of the unit circle.