§ 4

Prove that every subspace $\mathcal{L}^{\prime }$ lying in $\operatorname {Ker} \mathrm{A}$, and every subspace $\mathcal{L}^{\prime \prime }$ containing $\operatorname {Im} \mathrm{A}$, are invariant with respect to $\mathrm{A}$.

Prove that the sum and the intersection of invariant subspaces are invariant.

Let $\mathrm{A}$ be a linear transformation of a real space, $\lambda$ a complex root of its characteristic polynomial, $p=(\lambda +\bar{\lambda }) / 2$ and $q=\lambda \bar{\lambda }$. Prove that the dimension of the subspace $\mathcal{L}^{\prime }=\operatorname {Ker}\left(\mathrm{A}^{2}-p A+q E\right)$ is an even number.

Let $\mathrm{A}: \mathcal{L} \rightarrow \mathcal{L}$. Prove that $\mathcal{L}=\operatorname {Ker} \mathrm{A} \oplus \operatorname {Im} \mathrm{A}$ if and only if $\operatorname {Ker} \mathrm{A}^{2}=\operatorname {Ker} \mathrm{A}$.

Let $\mathcal{L}=\operatorname {Ker} \mathrm{A} \oplus \operatorname {Im} \mathrm{A}$. What form does the matrix of the transformation $\mathrm{A}$ have in the basis $\mathbf{e}$, if $e_{1}, \ldots , e_{r} \in \operatorname {Im} \mathrm{~ A}$, and $e_{r+1}, \ldots \ldots , e_{n} \in \operatorname {Ker} A$ ?

In some basis a transformation is given by the matrix

Does there exist a basis in which its matrix is diagonal? If yes, then find this basis.

Find the eigenvalues and the eigenspaces of the transformation given by the matrix

To every square matrix of order $n$ its transposed matrix is assigned. This defines a transformation $\mathrm{T}$ of the space of matrices. Find its eigenvalues and eigenspaces. Prove from these considerations that every matrix is uniquely represented as a sum of a symmetric and a skew-symmetric matrix.

Let $\mathrm{A}$ be a rotation of three-dimensional vector space about some axis through the angle $\alpha$ and let $A$ be the matrix of this rotation in some basis. Express $\alpha$ through the entries of $A$.

Let $\mathrm{A}$ be diagonalizable. Prove that the restriction of $\mathrm{A}$ to any invariant subspace is also diagonalizable.

The transformation $\mathrm{A}$ is given by the matrix

Find the change-of-basis matrix to some basis in which its matrix is upper triangular, and also the matrix of the transformation in this basis.

Let the matrix $A$ be nonsingular. Represent the matrix $A^{-1}$ as a polynomial in $A$.

Let a transformation $\mathrm{A}$ satisfy the relation $\mathrm{A}^{2}=\mathrm{A}$. Prove that $\mathrm{Rg} \mathrm{A}=\operatorname {tr} \mathrm{A}$.

Let a linear transformation $\mathrm{A}$ be given in some basis of an $n$-dimensional space $\mathcal{L}$ by a matrix $A$, and let an $(n-1)$-dimensional subspace $\mathcal{L}^{\prime }$ be given by the equation $\mathbf{u}^{T} \mathbf{x}=0$. Prove that $\mathcal{L}^{\prime }$ is invariant with respect to $\mathrm{A}$ if and only if there exists a factor $\lambda$ such that $A^{T} \mathbf{u}=\lambda \mathbf{u}$.