Eigenvalue Perturbation Theory

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Problem 4.1.24 from de Souza P.N., Silva J. "Berkeley Problems in Mathematics" [deSi04]More problems in Section 4.1 "Topology of $\reals[n]$"

Let $M_{n \times n}$ be the vector space of real $n \times n$ matrices, identified with $\mathbb {R}^{n^2}$ . Let $X \subset M_{n \times n}$ be a compact set. Let $S \subset \mathbb {C}$ be the set of all numbers that are eigenvalues of at least one element of $X$ . Prove that $S$ is compact.

Problem 7.5.6 from de Souza P.N., Silva J. "Berkeley Problems in Mathematics" [deSi04]More problems in Section 7.5 "Eigenvalues and Eigenvectors"

(1)(2)

Let $L$ be a real symmetric $n \times n$ matrix with $0$ as a simple eigenvalue, and let $v \in \mathbb {R}^{n}$ .

(1)

Show that for sufficiently small positive real $\varepsilon$ , the equation $Lx+\varepsilon x = v$ has a unique solution $x = x(\varepsilon ) \in \mathbb {R}^{n}$ .

(2)

Evaluate $\lim_{\varepsilon \to 0^{+}} \varepsilon x(\varepsilon )$ in terms of $v$ , the eigenvectors of $L$ , and the inner product $\left\langle \cdot , \cdot \right\rangle$ on $\mathbb {R}^{n}$ .

Problem VI.4.8 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section VI.4 "Asymptotics of Implicit Functions and Recurrent Sequences"

Find the derivatives with respect to $\varepsilon$ of the lengths of the semi-axes of the ellipsoid $x^{2}+y^{2}+z^{2}+x y+y z+z x=1+\varepsilon x y$ at $\varepsilon =0$ .

Problem 5.2.5 from Steeb W., Hardy Y. "Problems and Solutions in Introductory and Advanced Matrix Calculus" [StHa17]More problems in Section 5.2 "Supplementary Problems"

(i)(ii)

Let $\epsilon \in \mathbb {R}$ .

(i)

Find the eigenvalues and eigenvectors of the matrices

\begin{pmatrix} 1 & \epsilon \\ 1 & 1 \end{pmatrix}, \qquad \begin{pmatrix} 1 & 0 & \epsilon \\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{pmatrix}, \qquad \begin{pmatrix} 1 & 0 & 0 & \epsilon \\ 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 \end{pmatrix}.

Extend to $n \times n$ matrices.

(ii)

Find the eigenvalues and eigenvectors of the matrices

\begin{pmatrix} 0 & 1 \\ \epsilon & 0 \end{pmatrix}, \qquad \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ \epsilon & 0 & 0 \end{pmatrix}, \qquad \begin{pmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \epsilon & 0 & 0 & 0 \end{pmatrix}.

Extend to the $n \times n$ case.

References

[deSi04] de Souza P.N., Silva J. Berkeley Problems in Mathematics. 3rd ed., Springer, 2004.

[МГЛ+04] Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. Избранные задачи по вещественному анализу. 2-е изд., Невский Диалект; БХВ-Петербург, 2004.

[StHa17] Steeb W., Hardy Y. Problems and Solutions in Introductory and Advanced Matrix Calculus. 2nd ed., World Scientific Publishing, 2017.