Math Problems Vault

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.

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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.

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Remark.

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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.

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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).

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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).

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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.

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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.

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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 }.

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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).

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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].)

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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.

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