Self-adjoint and orthogonal transformations

Can the matrix of a self-adjoint transformation, in any basis whatsoever, be:

Prove that a matrix $A$ is the matrix of a self-adjoint transformation in a basis $e$ if and only if the matrix $\Gamma_{\mathbf{e}} A$ is symmetric.

Prove that all eigenvalues of a self-adjoint transformation are zero if and only if it is the zero transformation.

Find all self-adjoint nilpotent transformations.

Find all self-adjoint orthogonal transformations.

Find all self-adjoint idempotent transformations.

Let $\varphi$ be a self-adjoint transformation of the space $\mathcal{E}$. Prove that:

A transformation $\varphi$ is given in an orthonormal basis by the matrix $\mathbf{A}_{452}$. Find some orthonormal basis whose vectors lie in $\operatorname {Im} \varphi$ and $\operatorname {Ker} \varphi$, and the matrix of the transformation in this basis. $\mathbf{A}_{452} = \left[\begin{smallmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 3 \\ 1 & 2 & 3 & 4 \end{smallmatrix}\right]$.

How many orthonormal bases of eigenvectors exist for a given self-adjoint transformation if it:

Can a self-adjoint transformation have a basis of eigenvectors that is:

Let there exist in the space an orthonormal basis of eigenvectors of the transformation $\varphi$. Prove that the transformation $\varphi$ is self-adjoint.

Let $\lambda_{1}, \ldots , \lambda_{n}$ be the eigenvalues, and let $e_{1}, \ldots \ldots , e_{n}$ be an orthonormal basis of eigenvectors of a self-adjoint transformation $\varphi$ ($e_{2}$ belongs to $\lambda_{2}$). Find $\varphi (x)$ for an arbitrary vector $x$.

Prove that a matrix $A$ can be the matrix of a self-adjoint transformation $\varphi$ in some basis if and only if there exists a nondegenerate matrix $S$ such that $S^{ - 1} AS$ is a diagonal matrix.

Can a self-adjoint transformation, in any basis whatsoever, have the matrix:

Let $\mathcal{E} = \mathcal{L}_{1} \oplus \mathcal{L}_{2}$. Prove that the projection onto $\mathcal{L}_{1}$ parallel to $\mathcal{L}_{2}$ is a self-adjoint transformation if and only if $\mathcal{L}_{2} = \mathcal{L}_{1}^{\perp }$.

Let the transformations $\varphi$ and $\psi$ be self-adjoint. Prove that the following transformations will also be self-adjoint:

Prove that the product of self-adjoint transformations is self-adjoint if and only if they commute.

Prove that in every invariant subspace of a self-adjoint transformation there exists an orthonormal basis of eigenvectors of this transformation.

Find the transition matrix to an orthonormal basis of eigenvectors of the transformation $\varphi$ and the matrix of the transformation in this basis, if $\varphi$ is given in an orthonormal basis by the matrix:

A transformation $\varphi$ of an arithmetic space with the standard scalar product is given by the matrix $\mathbf{A}_{627}$. Find the eigenvalues and an orthonormal basis of eigenvectors of this transformation. $\mathbf{A}_{627} = \left[\begin{smallmatrix} x & y & y & \ldots & y \\ y & x & y & \ldots & y \\ \ldots & \ldots & \ldots & \ldots & . \\ y & y & y & \ldots & x \end{smallmatrix}\right]$.

Prove that for a self-adjoint transformation the dimension of the eigenspace corresponding to an eigenvalue $\lambda$ equals the multiplicity of $\lambda$ as a root of the characteristic polynomial.

Prove that two self-adjoint transformations commute if and only if they have a common orthonormal basis of eigenvectors.

For two self-adjoint transformations given in an orthonormal basis by the matrices:

find the transition matrix to a common orthonormal basis of eigenvectors and the matrices of the transformations in this basis.

Prove that any self-adjoint transformation of rank $r$ can be decomposed into a sum of $r$ self-adjoint transformations of rank 1.

Prove that any self-adjoint transformation can be decomposed into a linear combination of orthogonal projections onto pairwise orthogonal one-dimensional subspaces.

A self-adjoint transformation $\varphi$ is given in an orthonormal basis by the matrix $A$. Decompose $\varphi$ into a linear combination of orthogonal projections onto pairwise orthogonal subspaces:

Prove that self-adjoint transformations $\varphi$ and $\psi$ are equal if and only if $(\varphi (x), x) = (\psi (x), x)$ for any $x\in \mathcal{E}$.

Prove that a self-adjoint transformation $\varphi$ is positive (nonnegative) if and only if all its eigenvalues are positive (respectively, nonnegative).

Prove that a positive self-adjoint transformation can be decomposed into a product of $n$ contractions along pairwise orthogonal directions.

Decompose into a product of three contractions along pairwise orthogonal directions (alternatively, with respect to three pairwise orthogonal planes) the transformations given in an orthonormal basis by the matrices

Prove that a nonnegative self-adjoint transformation has an inverse if and only if it is positive.

A transformation $\varphi$ is given in an orthonormal basis by its matrix. Find the matrix of a positive self-adjoint transformation $\psi$ such that $\psi^{2} = \varphi$:

Let the eigenvalues of a self-adjoint transformation $\varphi$ be numbered so that $\lambda_{1} \geqslant \lambda_{2} \geqslant \ldots \geqslant \lambda_{n}$.

Prove that the diagonal elements of a symmetric matrix lie between its minimal and maximal characteristic numbers.

Let $A$ be a symmetric matrix, $\lambda_{1}$ and $\lambda_{n}$ its maximal and minimal characteristic numbers, and $\mu_{1}$ and $\mu_{k}$ the maximal and minimal characteristic numbers of its diagonal submatrix${ }^{1}$). Prove that $\lambda_{1} \geqslant \mu_{1} \geqslant \mu_{k} \geqslant \lambda_{n}$.

Let $\varphi$ be a linear transformation of a Euclidean space. Prove that:

Let the transformation $\varphi$ be isometric, i.e., $(\varphi (x), \varphi (y)) = (x, y)$ for any vectors $x$ and $y$. Prove that $\varphi$ is linear and one-to-one.

Prove that the orthogonal transformations of a Euclidean space $\mathcal{E}$ form a group with respect to the usual operation of composition of transformations.

Prove that for any two vectors of equal length there exists an orthogonal transformation carrying the first vector into the second.

Prove that for any two orthonormal bases there exists an orthogonal transformation carrying the first basis into the second.

Prove that the transformation of problem 28. 23 is orthogonal if and only if the matrix $A$ is orthogonal.

In a Euclidean space a basis with Gram matrix $\Gamma$ is chosen. Find the condition on the matrix of a linear transformation, necessary and sufficient for this transformation to be orthogonal.

Can an orthogonal transformation, in some basis, have the matrix:

An orthogonal transformation of an arithmetic space with the standard scalar product carries the columns of the matrix $A$ into the columns of $B$. How are the matrices $A$ and $B$ related?

A linear transformation $\varphi$ of an arithmetic space with the standard scalar product carries the columns of the matrix $A$ into the columns of the matrix $B$. Is $\varphi$ orthogonal:

Let $x_{1}, \ldots , x_{k}$ and $y_{1}, \ldots , y_{k}$ be vectors of an $n$-dimensional Euclidean space, and let an orthogonal transformation $\varphi$ be such that $\varphi \left(x_{i}\right) = y_{i}, \quad i = 1, \ldots , k$. Prove that this is possible if and only if the matrices of pairwise scalar products of the two systems of vectors are equal.

Let $\mathcal{L}$ be an invariant subspace of an orthogonal transformation $\varphi$. Prove that $\mathcal{L}^{\perp }$ is also invariant under $\varphi$. How is this result related to problem $25. 55?$

An orthogonal transformation is given in an orthonormal basis by the matrix $A$. Find the transition matrix $S$ to the canonical basis and the matrix $A^{\prime }$ of the transformation in this basis:

Prove that the transformation $\varphi^{ * } \varphi$ is orthogonal if and only if $\varphi$ is orthogonal.

Why is the equality

Find the polar decomposition of the matrix:

In the space $\mathcal{P}^{2}$ of polynomials of degree at most 2 with the standard scalar product, a transformation $\delta$ assigns to a polynomial its derivative. For the polar decomposition of $\delta$, write in the basis $1, t, t^{2}$ the matrix $Q$ of the orthogonal transformation and the matrix $V$ of the self-adjoint transformation.

Prove that for an arbitrary linear transformation $\varphi$ there exists a second form of the polar decomposition: $\varphi = \psi \theta$, where $\psi$ is a nonnegative self-adjoint transformation, and $\theta$ is orthogonal.

Prove that for a square matrix $A$ there exist orthogonal matrices $Q$ and $P$ such that $A = QDP$, where $D$ is a diagonal matrix with the singular values of the matrix $A$ on the diagonal.

Let $\varphi$ be a nondegenerate transformation, and $\varphi = \theta \psi = \psi_{1} \theta_{1}$, where $\psi , \psi_{1}$ are nonnegative self-adjoint, and $\theta , \theta_{1}$ are orthogonal transformations.

Prove that the nonnegative self-adjoint transformation $\psi$ in the polar decomposition $\varphi = \theta \psi$ is uniquely determined.

Prove that for a nondegenerate transformation the polar decomposition is unique.

Prove that a linear transformation is normal if and only if the factors in its polar decomposition commute.

Prove that the singular values of a self-adjoint transformation equal the moduli of its eigenvalues.

Find the singular values of an orthogonal matrix.

Let $\alpha_{1}, \ldots , \alpha_{n}$ be the singular values of a nondegenerate matrix $A$. Find the singular values of $A^{ - 1}$.

A matrix is multiplied by a number $\alpha$. How do its singular values change?

Prove that the singular values of a transformation $\varphi$ and its adjoint $\varphi^{ * }$ coincide.

Prove that the singular values of matrices $A$ and $B$ coincide if and only if there exist orthogonal matrices $U$ and $V$ such that $B = UAV$.

Prove that for a linear transformation $\varphi$ the ratio $\left|\varphi (x)\right| / \left|x\right|$, for any nonzero vector $x$, lies between the minimal and maximal singular value of $\varphi$.

Prove that the moduli of all eigenvalues of a transformation $\varphi$ belong to the interval $\left[\alpha_{n}, \alpha_{1}\right]$, where $\alpha_{1}$ and $\alpha_{n}$ are its largest and smallest singular values.

Prove that for a square matrix $A$ the product of the singular values equals $\left|\operatorname {det} A\right|$.

Find the singular values of the following matrices: