Math Problems Vault

Problem 45.1

Prove that the product of two self-adjoint operators in a metric vector space is a self-adjoint operator if and only if these operators commute.

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Problem 45.2

(a)(b)

Prove that if $\mathcal{A}$ and $\mathcal{B}$ are self-adjoint operators in a metric vector space, then:

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(a)

the operator $\mathcal{A B}+\mathcal{B} \mathcal{A}$ is self-adjoint;

(b)

for $\bar{\lambda }=-\lambda$ the operator $\lambda (\mathcal{A B}-\mathcal{B A})$ is self-adjoint.

Problem 45.3

Prove that the projection of a metric space $L_{1} \oplus L_{2}$ onto a subspace $L_{1}$ parallel to $L_{2}$ is a self-adjoint operator if and only if $L_{1}$ and $L_{2}$ are orthogonal.

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Problem 45.4

Find an eigenbasis (orthonormal) and the matrix in this basis of the operator given, in some orthonormal basis, by the matrix:

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(a)

$\left(\begin{array}{ll}2 & 1 \\ 1 & 2\end{array}\right) ; \quad$

(b)

$\left(\begin{array}{rrr}11 & 2 & -8 \\ 2 & 2 & 10 \\ -8 & 10 & 5\end{array}\right) ;$

(c)

$\left(\begin{array}{rrr}17 & -8 & 4 \\ -8 & 17 & -4 \\ 4 & -4 & 11\end{array}\right) ; \quad$

(d)

$\left(\begin{array}{rrr}5 & -1 & -1 \\ -1 & 5 & -1 \\ -1 & -1 & 5\end{array}\right) ;$

(e)

$\left(\begin{array}{lll}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{array}\right) ; \quad$

(f)

$\left(\begin{array}{llll}0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0\end{array}\right) ;$

(g)

$\left(\begin{array}{rrrr}1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & 1 & -1 \\ 1 & -1 & -1 & 1\end{array}\right)$ .

Problem 45.5

Prove that the functions

\frac{1}{\sqrt{2}}, \cos x, \sin x, \ldots , \cos n x, \sin n x

form an orthonormal eigenbasis for the symmetric operator $\frac{d^{2}}{d x^{2}}$ in the space

V_{n}=\left\{ a_{0}+a_{1} \cos x+b_{1} \sin x+\ldots +a_{n} \cos n x+b_{n} \sin n x \mid \quad a_{i}, b_{i} \in \mathbb {R}\right\}

with inner product $\frac{1}{\pi } \int_{-\pi }^{\pi } f(x) g(x) d x$ .

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Problem 45.6

Prove that the Legendre polynomials (problem 43.44) form an eigenbasis for the self-adjoint operator defined by the rule

(\mathcal{A}(f))(x)=\left(x^{2}-1\right) f^{\prime \prime }(x)+2 x f^{\prime }(x)

in the space of polynomials of degree $\leqslant n$ with inner product $\int_{-1}^{1} f(x) g(x) d x$ .

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Problem 45.7

(a)(b)(c)

Find an eigenbasis (orthonormal) and the matrix in this basis of the Hermitian operator given, in some orthonormal basis, by the matrix:

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(a)

$\left(\begin{array}{cc}3 & 2+2 i \\ 2-2 i & 1\end{array}\right)$ ;

(b)

$\left(\begin{array}{cc}3 & -i \\ i & 3\end{array}\right)$ ;

(c)

$\left(\begin{array}{cc}3 & 2-i \\ 2+i & 7\end{array}\right)$ .

Problem 45.8

(a)(b)(c)(d)

In the space of matrices $\mathbf{M}_{n}(\mathbb {C})$ set

(A, B)=\operatorname {tr}\left(A \cdot { }^{t} \bar{B}\right)

Prove that:

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(a)

$\mathbf{M}_{n}(\mathbb {C})$ is a Hermitian space;

(b)

every unitary matrix in this space has length $\sqrt{n}$ ;

(c)

the operators $X \mapsto A X$ and $X \mapsto { }^{t} \bar{A} X$ in the space $\mathbf{M}_{n}(\mathbb {C})$ are adjoint;

(d)

the operator $X \mapsto A X$ , where $A$ is a unitary matrix, is unitary.

Problem 45.9

Prove that self-adjoint operators of a Euclidean or Hermitian space commute if and only if they have a common orthonormal eigenbasis.

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Problem 45.10

(a)(b)

Prove that a self-adjoint linear operator in a Euclidean or Hermitian space:

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(a)

is nonnegative if and only if all its eigenvalues are nonnegative;

(b)

is positive if and only if all its eigenvalues are positive.

Problem 45.11

Prove that if $\mathcal{A}$ is an operator in a Euclidean or Hermitian space, then $\mathcal{A}^{*} \mathcal{A}$ is a nonnegative self-adjoint operator, and it is positive if and only if the operator $\mathcal{A}$ is invertible.

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Problem 45.12

Prove that if two nonnegative self-adjoint operators in a Euclidean or Hermitian space commute, then their product is a nonnegative self-adjoint operator.

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Problem 45.13

Prove that for every nonnegative (positive) self-adjoint operator $\mathcal{A}$ in a Euclidean or Hermitian space there exists a nonnegative (positive) self-adjoint operator $\mathcal{B}$ such that $\mathcal{B}^{2}=\mathcal{A}$ .

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Problem 45.14

Let the operator $\mathcal{A}$ in a three-dimensional Euclidean space be given, in some orthonormal basis, by the matrix

\left[\begin{smallmatrix} 13 & 14 & 4 \\ 14 & 24 & 18 \\ 4 & 18 & 29 \end{smallmatrix}\right]

Find, in this basis, the matrix of the positive self-adjoint operator $\mathcal{B}$ such that $\mathcal{B}^{2}=\mathcal{A}$ .

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Problem 45.15

Prove that the eigenvalues of the product of two nonnegative self-adjoint operators in a Euclidean or Hermitian space, one of which is invertible, are real and nonnegative.

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Problem 45.16

Prove that a nonnegative self-adjoint operator of rank $r$ in a Euclidean or Hermitian space is the sum of $r$ nonnegative self-adjoint operators of rank 1.

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Problem 45.17

Prove that every linear operator $\mathcal{A}$ of a Hermitian space is uniquely representable in the form

\mathcal{A}=\mathcal{A}_{1}+i \mathcal{A}_{2}

where $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ are Hermitian operators.

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Problem 45.18

Let $A$ be a real Jacobi matrix, i.e., a symmetric matrix of the form

\left[\begin{smallmatrix} \alpha _{1} & \beta _{1} & 0 & \ldots & 0 \\ \beta _{1} & \alpha _{2} & \beta _{2} & \ldots & 0 \\ 0 & \beta _{2} & \alpha _{3} & \ldots & 0 \\ \ldots \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & \beta _{n-1} \\ 0 & 0 & 0 & \ldots & \alpha _{n} \end{smallmatrix}\right]

with $\beta_{1} \cdot \ldots \cdot \beta_{n-1} \neq 0$ . Prove that $A$ has no repeated eigenvalues.

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Problem 45.19

Find the orthogonal transformation reducing the quadratic function to principal axes:

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(a)

$6 x_{1}^{2}+5 x_{2}^{2}+7 x_{3}^{2}-4 x_{1} x_{2}+4 x_{1} x_{3}$ ;

(b)

$11 x_{1}^{2}+5 x_{2}^{2}+2 x_{3}^{2}+16 x_{1} x_{2}+4 x_{1} x_{3}-20 x_{2} x_{3}$ ;

(c)

$x_{1}^{2}+x_{2}^{2}+5 x_{3}^{2}-6 x_{1} x_{2}-2 x_{1} x_{3}+2 x_{2} x_{3}$ ;

(d)

$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+4 x_{1} x_{2}+4 x_{1} x_{3}+4 x_{2} x_{3}$ ;

(e)

$x_{1}^{2}-5 x_{2}^{2}+x_{3}^{2}+4 x_{1} x_{2}+2 x_{1} x_{3}+4 x_{2} x_{3}$ ;

(f)

$2 x_{1} x_{2}-6 x_{1} x_{3}-6 x_{2} x_{4}+2 x_{3} x_{4}$ ;

(g)

$3 x_{1}^{2}+8 x_{1} x_{2}-3 x_{2}^{2}+4 x_{3}^{2}-4 x_{3} x_{4}+x_{4}^{2}$ ;

(h)

$x_{1}^{2}+2 x_{1} x_{2}+x_{2}^{2}-2 x_{3}^{2}-4 x_{3} x_{4}-2 x_{4}^{2}$ ;

(i)

$9 x_{1}^{2}+5 x_{2}^{2}+5 x_{3}^{2}+8 x_{4}^{2}+8 x_{2} x_{3}-4 x_{2} x_{4}+4 x_{3} x_{4}$ ;

(j)

$4 x_{1}^{2}-4 x_{2}^{2}-8 x_{2} x_{3}+2 x_{3}^{2}-5 x_{4}^{2}+6 x_{4} x_{5}+3 x_{5}^{2}$ .

Problem 45.20

Prove that if $f(x)=\sum_{i=1}^{r} \lambda_{i} x_{i}^{2}$ , then

\max \left(\left|\lambda _{1}\right|, \ldots ,\left|\lambda _{r}\right|\right)=\max _{\left|x\right|=1}\left|f(x)\right|

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Problem 45.21

(a)(b)(c)

Reduce the Hermitian quadratic function to principal axes:

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(a)

$5\left|x_{1}\right|^{2}+i \sqrt{3} x_{1} \bar{x}_{2}-i \sqrt{3} \bar{x}_{1} x_{2}+6\left|x_{2}\right|^{2}$ ;

(b)

$2\left|x_{1}\right|^{2}+\left|x_{2}\right|^{2}+2 i x_{1} \bar{x}_{2}-2 i \bar{x}_{1} x_{2}+2 i \bar{x}_{2} x_{3}-2 i x_{2} \bar{x}_{3}$ ;

(c)

$\left|x_{1}\right|^{2}+2\left|x_{2}\right|^{2}+3\left|x_{3}\right|^{2}-2 \bar{x}_{1} x_{2}+2 i x_{1} \bar{x}_{2}+2 i \bar{x}_{2} x_{3}-2 i x_{2} \bar{x}_{3}$ .