Other problems (24.139-24.148)

Let the linear transformations $\varphi , \psi$ commute. Prove that:

Let $\lambda_{0}$ be an eigenvalue of a linear transformation $\varphi$.

Prove that:

Let $\varphi$ be a singular linear transformation of a finite-dimensional vector space. Prove that there exists $\varepsilon_{0} > 0$ such that for all $\left|\varepsilon \right| \leqslant \varepsilon_{0}$, the transformation $\varphi + \varepsilon \iota$ is nonsingular.

Prove that for any two linear transformations $\varphi , \psi$ of the same vector space, the characteristic polynomials of the transformations $\varphi \psi$ and $\psi \varphi$ coincide.

Let $\varphi , \psi$ be commuting linear transformations of an $n$-dimensional space, and let $\varphi$ have $n$ distinct eigenvalues. Prove that all eigenvectors of the transformation $\varphi$ are also eigenvectors for $\psi$, so that the matrices of $\varphi$ and $\psi$ are diagonal in a common basis for them.

Let a linear transformation $\varphi$ be diagonalizable and let each of its eigenspaces be invariant with respect to a linear transformation $\psi$. Prove that $\varphi \psi = \psi \varphi$.

Let $\mathcal{L} = \mathcal{L}^{\prime } \oplus \mathcal{L}^{\prime \prime }$, where $\mathcal{L}^{\prime }, \mathcal{L}^{\prime \prime }$ are nonzero linear subspaces in $\mathcal{L}$.

Let $\varphi , \psi$ be linear transformations of an $n$-dimensional vector space. It is given that $\varphi^{n} = \theta , \operatorname {dim} \operatorname {Ker} \varphi = 1$ and $\left[\psi , \varphi \right] = \psi \varphi - \varphi \psi = \varphi$. Prove that $\psi$ has $n$ eigenvalues of the form $\lambda , \lambda - 1, \ldots , \lambda - (n - 1)$, where $\lambda$ is some number.

Let $\varphi$ and $\psi$ be linear transformations of the space $\mathcal{L}$, and let $\varphi$ be one-to-one. Find $\varepsilon > 0$ such that for any $\delta \in (- \varepsilon , \varepsilon )$ the linear transformation $\varphi + \delta \psi$ is one-to-one.