Canonical Forms

Let $A$ be a complex $n \times n$ matrix that has finite order; that is, $A^{k} = I$ for some positive integer $k$. Prove that $A$ is diagonalizable.

Prove, or supply a counterexample: If $A$ is an invertible $n \times n$ complex matrix and some power of $A$ is diagonal, then $A$ can be diagonalized.

Let

Express $A^{-1}$ as a polynomial in $A$ with real coefficients.

For $x \in \mathbb {R}$, let

Compute $A^{10}$ for the matrix

Calculate $A^{100}$ and $A^{-7}$, where

Prove or disprove: For any $2 \times 2$ matrix $A$ over $\mathbb {C}$, there is a $2 \times 2$ matrix $B$ such that $A = B^{2}$.

Is there a real $2 \times 2$ matrix $A$ such that

Exhibit such an $A$ or prove there is none.

For which positive integers $n$ is there a $2 \times 2$ matrix

with integer entries and order $n$; that is, $A^{n} = I$ but $A^{k} \neq I$ for $0 < k < n$?

See also Problem 7.7.8.

Find a square root of the matrix

How many square roots does this matrix have?

Let $A$ denote the matrix

For which positive integers $n$ is there a complex $4 \times 4$ matrix $X$ such that $X^{n} = A$?

Prove or disprove: There is a real $n \times n$ matrix $A$ such that

if and only if $n$ is even.

Let $A$ be an $n \times n$ Hermitian matrix satisfying the condition

Show that $A = I$.

Which of the following matrix equations have a real matrix solution $X$? (It is not necessary to exhibit solutions.)

Find a real matrix $B$ such that

Let $V$ be a finite-dimensional vector space and $T: V \to V$ a diagonalizable linear transformation. Let $W \subset V$ be a linear subspace which is mapped into itself by $T$. Show that the restriction of $T$ to $W$ is diagonalizable.

Let $A$ and $B$ be diagonalizable linear transformations of $\mathbb {R}^{n}$ into itself such that $AB = BA$. Let $E$ be an eigenspace of $A$. Prove that the restriction of $B$ to $E$ is diagonalizable.

Let $V$ be a finite-dimensional complex vector space and let $A$ and $B$ be linear operators on $V$ such that $AB = BA$. Prove that if $A$ and $B$ can each be diagonalized, then there is a basis for $V$ which simultaneously diagonalizes $A$ and $B$.

Let $A$ and $B$ be $n \times n$ complex matrices. Prove or disprove each of the following statements:

Let

Find a real matrix $B$ such that $B^{-1}AB$ is diagonal.

Let $A: \mathbb {R}^{6} \to \mathbb {R}^{6}$ be a linear transformation such that $A^{26} = I$. Show that $\mathbb {R}^{6} = V_{1} \oplus V_{2} \oplus V_{3}$, where $V_{1}$, $V_{2}$, and $V_{3}$ are two-dimensional invariant subspaces for $A$.

Determine the Jordan Canonical Form of the matrix

Find the eigenvalues, eigenvectors, and the Jordan Canonical Form of

considered as a matrix with entries in $\mathbf{F}_{3} = \mathbb {Z}/3\mathbb {Z}$.

Let $A$ be an $n \times n$ complex matrix, and let $\chi$ and $\mu$ be the characteristic and minimal polynomials of $A$. Suppose that

Determine the Jordan Canonical Form of $A$.

Let $M$ be the $n \times n$ matrix over a field $\mathbf{F}$, all of whose entries are equal to $1$.

Let $M_{2 \times 2}$ denote the vector space of complex $2 \times 2$ matrices. Let

and let the linear transformation $T: M_{2 \times 2} \to M_{2 \times 2}$ be defined by $T(X) = XA-AX$. Find the Jordan Canonical Form for $T$.

Let $M_{n}$ be the vector space of $n \times n$ complex matrices. For $A$ in $M_{n}$ define the linear transformation $T_{A}$ on $M_{n}$ by $T_{A}(X) = AX-XA$. Prove that the rank of $T_{A}$ is at most $n^{2}-n$.

Let $A$ be an $n \times n$ matrix with real entries, let $\xi (t)$ denote its characteristic polynomial, and let $g(t) \in \mathbb {R}^{t}$ be a polynomial of degree $n-1$ dividing $\xi (t)$. What are the possibilities for the rank of $g(A)$?

Find the Jordan Canonical Form of the matrix

Let $A$ and $B$ be complex matrices of sizes $3 \times 5$ and $5 \times 3$, respectively, such that

Find the Jordan canonical form of $BA$.

Let $A = (a_{ij})$ be a $n \times n$ complex matrix such that $a_{ij} \neq 0$ if $i=j+1$ but $a_{ij} = 0$ if $i \geq j+2$. Prove that $A$ cannot have more than one Jordan block for any eigenvalue.

Let $A$ be a real, upper-triangular, $n \times n$ matrix that commutes with its transpose. Prove that $A$ is diagonal.

Let $A$ be an $n \times n$ complex matrix. Prove there is a unitary matrix $U$ such that $B = UAU^{-1}$ is upper-triangular: $B_{jk} = 0$ for $j > k$.

Let $b$ be a real nonzero $n \times 1$ matrix (a column vector). Set $M = bb^{t}$ (an $n \times n$ matrix) where $b^{t}$ denotes the transpose of $b$.

Let $M$ be an invertible real $n \times n$ matrix. Show that there is a decomposition $M = UT$ in which $U$ is an $n \times n$ real orthogonal matrix and $T$ is upper-triangular with positive diagonal entries. Is this decomposition unique?

Let $A$ be a nonsingular real $n \times n$ matrix. Prove that there exists a unique orthogonal matrix $Q$ and a unique positive definite symmetric matrix $B$ such that $A = QB$.

Let $A$ be a $2 \times 2$ matrix with complex entries. Prove that the series $I+A+A^{2}+\ldots$ converges if and only if every eigenvalue of $A$ has absolute value less than $1$.

Let $A$ be the $3 \times 3$ matrix

Determine all real numbers $a$ for which the limit $\lim_{n \to \infty } a^{n}A^{n}$ exists and is nonzero (as a matrix).

Suppose $p$ is a prime. Show that every element of $GL_{2}(\mathbf{F}_{p})$ has order dividing either $p^{2}-1$ or $p(p-1)$.