Math Problems Vault

Chapter VII

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Section VII.1§ 1[7/100%]

Problem VII.1.1

(a)(b)(c)

Verify that in the space of polynomials of degree $\leqslant 2$ an inner product can be defined by the formula

(p, q)=\int _{-1}^{1} p(t) q(t) d t

(a)

Write the Gram matrix of the basis 1, $t, t^{2}$ .

(b)

Using the change-of-basis matrix, find the Gram matrix of the basis 1, $(t-1)$ , $(t-1)^{2}$ .

(c)

Find the angle between the polynomials $t^{2}+1$ and $t+1$ .

Problem VII.1.2

A subspace of a Euclidean space is given in an orthonormal basis by the equation $\xi^{1}+\xi^{2}+\xi^{3}+\xi^{4}=0$ . Find an orthonormal basis in this subspace.

Problem VII.1.3

(a)(b)

Let $\operatorname {dim} \mathcal{E}=4$ and let $\mathcal{E}^{\prime } \subseteq \mathcal{E}$ be given in an orthonormal basis by the system

\xi _{1}+\xi _{2}+\xi _{3}=0, \xi _{2}+\xi _{3}+\xi _{4}=0 .

Find:

(a)

a basis in $\mathcal{E}^{\prime \perp }$

(b)

the orthogonal projection onto $\mathcal{E}^{\prime }$ of the vector $a$ with coordinates $\mathbf{a}=\| 1,2,3,4\|^{T}$ .

Problem VII.1.4

(a)(b)

Suppose that all the entries of an orthogonal matrix of order $n$ are equal to one another in absolute value.

(a)

What is the absolute value of an entry of such a matrix?

(b)

Prove that such matrices exist if $n=2^{k}$ , where $k$ is a natural number.

Problem VII.1.5

(a)(b)

Find the $Q R$ decomposition of the matrix

(a)

$A=\left\| \begin{array}{ll}1 & 1 \\ 2 & 3\end{array}\right\|$

(b)

$B=\left\| \begin{array}{rrr}1 & 3 & 1 \\ 2 & 4 & -3 \\ 1 & 1 & -1\end{array}\right\|$ .

Problem VII.1.6

Prove that the $Q R$ decomposition of a matrix is unique.

Problem VII.1.7

In a four-dimensional Euclidean space a three-dimensional parallelepiped is built on the vectors having in an orthonormal basis the coordinate columns $\| 1,1,-1,0\|^{T}$ , $\| 1,1,1,-1\|^{T}$ and $\| 1,1,1,1\|^{T}$ . Find the volume of the parallelepiped.

Section VII.2§ 2[11/100%]

Problem VII.2.1

(a)(b)(c)

In a basis $e$ with Gram matrix $\Gamma$ the transformation $\mathrm{A}$ has matrix $A$ .

(a)

Find the matrix of the adjoint transformation. Find the eigenspaces

(b)

of the transformation $\mathrm{A}$

(c)

of the transformation $\mathrm{A}^{*}$

A=\left\| \begin{array}{rr} -1 & -2 \\ 3 & 4 \end{array}\right\| , \quad \Gamma =\left\| \begin{array}{ll} 1 & 1 \\ 1 & 2 \end{array}\right\| .

Problem VII.2.2

Let a transformation $\mathrm{A}$ of a real linear space $\mathcal{L}$ have a symmetric matrix in some basis $\mathbf{e}$ . Prove that it is diagonalizable.

Problem VII.2.3

Find all linear transformations which are both orthogonal and self-adjoint.

Problem VII.2.4

(a)(b)(c)

How many orthonormal bases of eigenvectors of a given self-adjoint transformation of an $n$ -dimensional space are there, if its characteristic polynomial

(a)

has no multiple roots,

(b)

has multiple roots?

(c)

Is a non-orthogonal basis of eigenvectors of a self-adjoint transformation possible?

Problem VII.2.5

Find the change-of-basis matrix $S$ to an orthonormal basis of eigenvectors of the transformation given in an orthonormal basis by the matrix $A$ , and write the matrix $A^{\prime }$ of the transformation in the basis found,

A=\left\| \begin{array}{lll} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{array}\right\| .

Problem VII.2.6

Decompose the orthogonal transformation given by the matrix

\left\| \begin{array}{rrrr} 0 & 0 & 0 & -1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right\|

in an orthonormal basis into a product of two rotations in mutually perpendicular two-dimensional subspaces.

Problem VII.2.7

Find the singular value decomposition of the matrix

A=\left\| \begin{array}{lll} 2 & 1 & 0 \\ 8 & 2 & 2 \\ 2 & 0 & 1 \end{array}\right\| .

Problem VII.2.8

(a)(b)

Obtain the polar decomposition of the transformation given in an orthonormal basis by the matrix

(a)

$A=\left\| \begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right\|$

(b)

$B=\left\| \begin{array}{cc}\sqrt{2} & 1 \\ 0 & \sqrt{2}\end{array}\right\|$ .

Problem VII.2.9

What are the singular values of a square matrix $A$ of order $n$ , if $\operatorname {Rg} A=1$ ?

Problem VII.2.10

Let $\alpha , \beta$ and $\gamma$ be the maximal singular values of the square matrices $A, B$ and $A B$ respectively. Prove that $\gamma \leqslant \alpha \beta$ .

Problem VII.2.11

Prove that all the roots of the characteristic polynomial of an orthogonal transformation have absolute value equal to one.

Section VII.3§ 3[9/100%]

Problem VII.3.1

In the space of polynomials of degree $\leqslant 3$ we define the scalar product in the same way as in problem 1 of § 1. A linear function $\mathrm{f}$ assigns to a polynomial $p(t)$ its constant term $p(0)$ . Find the vector (polynomial) associated with this linear function.

Problem VII.3.2

A linear transformation $\mathrm{A}$ is associated with a bilinear function $\mathrm{b}$ . With which bilinear function is its adjoint transformation $\mathrm{A}^{*}$ associated?

Problem VII.3.3

In a basis $\mathbf{e}$ a bilinear function has matrix $B$ . Find the matrix of its associated transformation if $\Gamma$ is the Gram matrix of the basis $\mathbf{e}$ :

B=\left\| \begin{array}{ll} 2 & 1 \\ 1 & 2 \end{array}\right\| , \quad \Gamma =\left\| \begin{array}{ll} 1 & 1 \\ 1 & 2 \end{array}\right\| .

Problem VII.3.4

Prove that the value of a quadratic form $\mathrm{k}(x)$ on a vector $x$ of length 1 lies between the smallest and the largest eigenvalues of its associated transformation, and that these bounds are attained on the corresponding eigenvectors.

Problem VII.3.5

A quadratic form is given in an orthonormal basis by the polynomial

3\left(\xi ^{1}\right)^{2}+3\left(\xi ^{2}\right)^{2}+3\left(\xi ^{3}\right)^{2}-2 \xi ^{1} \xi ^{2}-2 \xi ^{1} \xi ^{3}-2 \xi ^{2} \xi ^{3} .

Find the change-of-basis matrix to an orthonormal basis in which it has diagonal form, and its form in this basis.

Problem VII.3.6

Let $\mathrm{k}$ and $\mathrm{h}$ be quadratic forms and let $\mathrm{h}$ be positive definite. Does there exist a basis in which $\mathrm{k}$ has canonical form and $\mathrm{h}$ — diagonal form?

Problem VII.3.7

(a)(b)

Give an example of two quadratic forms which

(a)

cannot be brought to diagonal form in one and the same basis

(b)

can be brought to diagonal form in one and the same basis, but neither of which is positive definite or negative definite.

Problem VII.3.8

Find the change-of-basis matrix to a basis in which the quadratic forms $\mathrm{k}(x)=\left(\xi^{1}\right)^{2}-2 \xi^{1} \xi^{2}+\left(\xi^{2}\right)^{2}$ and $\mathrm{h}(x)=17\left(\xi^{1}\right)^{2}+8 \xi^{1} \xi^{2}+\left(\xi^{2}\right)^{2}$ both have diagonal form, and also their form in this basis.

Problem VII.3.9

Prove that, in order that for two non-proportional quadratic forms in a two-dimensional space there exist a basis in which they both have diagonal form, it is necessary and sufficient that among their linear combinations there be a positive definite form. How essential is the assumption about the dimension of the space here?

Section VII.4§ 4[5/100%]

Problem VII.4.1

(a)(b)

In a two-dimensional unitary space an orthonormal basis is given, together with vectors $a$ and $b$ whose coordinates in this basis are respectively $1+i, 1-i$ and $-i, 2-2 i$ .

(a)

Find their lengths, and also the cosines of the angles between $a$ and $b$ and between $b$ and $a$ .

(b)

Orthogonalize this pair of vectors.

Problem VII.4.2

Write down some non-real Hermitian matrix and some non-real unitary matrix of order 3.

Problem VII.4.3

Prove that the roots of the characteristic equation of a real orthogonal matrix (including the complex ones) are equal to 1 in modulus.

Problem VII.4.4

Find an orthonormal basis of eigenvectors and the matrix of the transformation in this basis for the transformation $\mathrm{A}$ given in an orthonormal basis by the matrix

\left\| \begin{array}{cc} 0 & i \\ -i & 0 \end{array}\right\| .

Is the transformation self-adjoint, unitary?

Problem VII.4.5

Find an orthonormal basis of eigenvectors of the unitary transformation given in an orthonormal basis by the matrix

A=\left\| \begin{array}{cr} \cos \varphi & -\sin \varphi \\ \sin \varphi & \cos \varphi \end{array}\right\| .