§ 2

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

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.

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

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

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,

Decompose the orthogonal transformation given by the matrix

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

Find the singular value decomposition of the matrix

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

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

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

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