Self-adjoint transformations (29.1-29.37)

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: