Eigenvalues and Eigenvectors

Let $M$ be a real $3 \times 3$ matrix such that $M^{3} = I$, $M \neq I$.

Let $N$ be a linear operator on an $n$-dimensional vector space, $n > 1$, such that $N^{n} = 0$, $N^{n-1} \neq 0$. Prove there is no operator $X$ with $X^{2} = N$.

Let $\mathbf{F}$ be a field, $n$ and $m$ positive integers, and $A$ an $n \times n$ matrix with entries in $\mathbf{F}$ such that $A^{m} = 0$. Prove that $A^{n} = 0$.

Let $B$ be a $3 \times 3$ matrix whose null space is $2$-dimensional, and let $\chi (\lambda )$ be the characteristic polynomial of $B$. For each assertion below, provide either a proof or a counterexample.

Suppose that the minimal polynomial of a linear operator $T$ on a seven-dimensional vector space is $x^{2}$. What are the possible values of the dimension of the kernel of $T$?

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

Let $V$ be a finite-dimensional vector space over the rationals $\mathbb {Q}$ and let $M$ be an automorphism of $V$ such that $M$ fixes no nonzero vector in $V$. Suppose that $M^{p}$ is the identity map on $V$, where $p$ is a prime number. Show that the dimension of $V$ is divisible by $p-1$.

Let $\mathbf{F}$ be a field, $V$ a finite-dimensional vector space over $\mathbf{F}$, and $T$ a linear transformation of $V$ into $V$ whose minimum polynomial, $\mu$, is irreducible over $\mathbf{F}$.

Suppose $A$ and $M$ are $n \times n$ matrices over $\mathbb {C}$, $A$ is invertible and $AMA^{-1} = M^{2}$. Prove the nonzero eigenvalues of $M$ are roots of unity.

Prove that the matrix

has two positive and two negative eigenvalues (counting multiplicities).

Prove that the matrix

has one positive eigenvalue and one negative eigenvalue.

For arbitrary elements $a$, $b$, and $c$ in a field $\mathbf{F}$, compute the minimal polynomial of the matrix

Let $A$ be an $n \times n$ complex matrix such that the three matrices $A+I$, $A^{2}+I$, $A^{3}+I$ are all unitary. Prove that $A$ is the zero matrix.

Let $k$ be a field, and let $n \geq 1$. Prove that the following properties of an $n \times n$ matrix $A$ with entries in $k$ are equivalent:

• $A$ is a scalar multiple of the identity matrix.

• Every nonzero vector $v \in k^{n}$ is an eigenvector of $A$.

Suppose that $A$ and $B$ are endomorphisms of a finite-dimensional vector space $V$ over a field $\mathbf{F}$. Prove or disprove the following statements:

Let $A$ and $B$ denote real $n \times n$ symmetric matrices such that $AB = BA$. Prove that $A$ and $B$ have a common eigenvector in $\mathbb {R}^{n}$.

Let $S$ be a nonempty commuting set of $n \times n$ complex matrices ($n \geq 1$). Prove that the members of $S$ have a common eigenvector.

Let $A$ and $B$ be complex $n \times n$ matrices such that $AB = BA^{2}$, and assume $A$ has no eigenvalues of absolute value $1$. Prove that $A$ and $B$ have a common (nonzero) eigenvector.

Let $V$ be a finite-dimensional vector space over an algebraically closed field. A linear operator $T : V \to V$ is called completely reducible if whenever a linear subspace $E \subset V$ is invariant under $T$, that is $T(E) \subset E$, there is a linear subspace $F \subset V$ which is invariant under $T$ and such that $V = E \oplus F$. Prove that $T$ is completely reducible if and only if $V$ has a basis of eigenvectors.

Let $V$ be the vector space of sequences $(a_{n})$ of complex numbers. The shift operator $S : V \to V$ is defined by

Let $T$ be a linear transformation on a finite-dimensional $\mathbb {C}$-vector space $V$, and let $f$ be a polynomial with coefficients in $\mathbb {C}$. If $\lambda$ is an eigenvalue of $T$, show that $f(\lambda )$ is an eigenvalue of $f(T)$. Is every eigenvalue of $f(T)$ necessarily obtained in this way?

Let $A$ be the $n \times n$ matrix which has zeros on the main diagonal and ones everywhere else. Find the eigenvalues and eigenspaces of $A$ and compute $\operatorname {det}\left(A\right)$.

Let $A$ and $B$ be two $n \times n$ self-adjoint (i.e., Hermitian) matrices over $\mathbb {C}$ and assume $A$ is positive definite. Prove that all eigenvalues of $AB$ are real.

Let $a$, $b$, $c$, and $d$ be real numbers, not all zero. Find the eigenvalues of the following $4 \times 4$ matrix and describe the eigenspace decomposition of $\mathbb {R}^{4}$:

Show that the following three conditions are all equivalent for a real $3 \times 3$ symmetric matrix $A$, whose eigenvalues are $\lambda_{1}$, $\lambda_{2}$, and $\lambda_{3}$:

Let

be a real matrix with $a, b, c, d > 0$. Show that $A$ has an eigenvector

with $x, y > 0$.

Let $n$ be a positive integer, and let $A = (a_{ij})_{i,j=1}^{n}$ be the $n \times n$ matrix with $a_{ii} = 2$, $a_{i,i\pm 1} = -1$, and $a_{ij} = 0$ otherwise; that is,

Prove that every eigenvalue of $A$ is a positive real number.

Let $A$ be a real symmetric $n \times n$ matrix with nonnegative entries. Prove that $A$ has an eigenvector with nonnegative entries.

Let $A_{n}$ be the $n \times n$ matrix whose entries $a_{jk}$ are given by

Prove that the eigenvalues of $A$ are symmetric with respect to the origin.

Let $A = (a_{ij})_{i,j=1}^{n}$ be a real $n \times n$ matrix with nonnegative entries such that

Prove that no eigenvalue of $A$ has absolute value greater than $1$.

Let $S$ be a special orthogonal (that is, orthogonal with determinant $1$) $n \times n$ matrix, with $n$ odd.

Let $A$ and $B$ be $m \times m$ Hermitian matrices such that all the eigenvalues of $A$ lie in $[a,a']$ and all the eigenvalues of $B$ lie in $[b,b']$. Show that all eigenvalues of $A+B$ lie in $[a+b,a'+b']$.

Let $A$ be an $n \times n$ real symmetric matrix with $a_{11} = a_{22} = \cdots = a_{nn} = 2$ and $a_{ij} \in \left\{ -1,0\right\}$ for $i \neq j$. Suppose $k$ of the off-diagonal entries in the first row and column are $-1$'s and the rest are $0$'s, and suppose that this is the maximum number of $-1$'s in any row or column. Show that all eigenvalues of $A$ lie in the interval $[k-1, k+1]$.

Let $A$ and $B$ be real symmetric $n \times n$ matrices, and assume $B$ is positive definite. Show that there is a nonzero vector $x$ that maximizes

and that any such $x$ is an eigenvector of the matrix $B^{-1}A$.

Let $A$ and $B$ be $n \times n$ real symmetric matrices, and suppose $A$ is positive definite. Prove that

exists for every $y \in \mathbb {R}^{n}$, $y \neq 0$.