General Theory of Matrices

Prove that a commutative $\mathbb {C}$-algebra of $2 \times 2$ complex matrices has dimension at most $2$ over $\mathbb {C}$.

Prove the following three statements about real $n \times n$ matrices.

Let $B$ denote the matrix

where $a$, $b$, and $c$ are real and $\left|a\right|$, $\left|b\right|$, and $\left|c\right|$ are distinct. Show that there are exactly four symmetric matrices of the form $BQ$, where $Q$ is a real orthogonal matrix of determinant $1$.

Let $P$ be a $n \times n$ real matrix such that $x^{t}Py = -y^{t}Px$ for all column vectors $x, y$ in $\mathbb {R}^{n}$. Prove that $P$ is skew-symmetric.

Let $A$ be a real skew-symmetric matrix ($A_{ij} = -A_{ji}$). Prove that $A$ has even rank.

A real symmetric $n \times n$ matrix $A$ is called positive semi-definite if $x^{t}Ax \geq 0$ for all $x \in \mathbb {R}^{n}$. Prove that $A$ is positive semi-definite if and only if $\operatorname {tr} AB \geq 0$ for every real symmetric positive semi-definite $n \times n$ matrix $B$.

Suppose that $A$ and $B$ are real matrices such that

for all $v \in \mathbb {R}^{n}$ and

Show that $AB = BA = 0$ and give an example where neither $A$ nor $B$ is zero.

Let $A$ and $B$ be $n \times n$ matrices over $\mathbb {R}$ such that $A+B$ is invertible. Prove that

Suppose $A$ is a real $n \times n$ matrix.

Let $M_{1} = \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix}$, $M_{2} = \begin{pmatrix} 5 & 7 \\ -3 & -4 \end{pmatrix}$, $M_{3} = \begin{pmatrix} 5 & 6.9 \\ -3 & -4 \end{pmatrix}$. For which (if any) $i$, $1 \leq i \leq 3$, is the sequence $(M_{i}^{n})$ bounded away from $\infty$? For which $i$ is the sequence bounded away from $0$?

Let $A$ be an $n \times n$ complex matrix, all of whose eigenvalues are equal to $1$. Suppose that the set $\left\{ A^{n} \mid n = 1, 2, \ldots \right\}$ is bounded. Show that $A$ is the identity matrix.

Consider the complex $3 \times 3$ matrix

where $a_{0}, a_{1}, a_{2} \in \mathbb {C}$.

Let $A$ be a $n \times n$ real matrix.

Let the real $2n \times 2n$ matrix $X$ have the form

where $A, B, C$, and $D$ are $n \times n$ matrices that commute with one another. Prove that $X$ is invertible if and only if $AD-BC$ is invertible.

Let $GL_{2}(\mathbb {C})$ denote the group of invertible $2 \times 2$ matrices with coefficients in the field of complex numbers. Let $PGL_{2}(\mathbb {C})$ denote the quotient of $GL_{2}(\mathbb {C})$ by the normal subgroup $\left\{ \begin{pmatrix} \lambda & 0 \\ 0 & \lambda \end{pmatrix} \mid \lambda \in \mathbb {C}^{*}\right\}$. Let $n$ be a positive integer, and suppose that $a, b$ are elements of $PGL_{2}(\mathbb {C})$ of order exactly $n$. Prove that there exists $c \in PGL_{2}(\mathbb {C})$ such that $cac^{-1}$ is a power of $b$.

Let $B = (b_{ij})_{i,j=1}^{20}$ be a real $20 \times 20$ matrix such that

Prove that $B$ is nonsingular.

Let

Show that every real matrix $B$ such that $AB = BA$ has the form $sI+tA$, where $s, t \in \mathbb {R}$.

Let $A$ be a $2 \times 2$ matrix over $\mathbb {C}$ which is not a scalar multiple of the identity matrix $I$. Show that any $2 \times 2$ matrix $X$ over $\mathbb {C}$ commuting with $A$ has the form $X = \alpha I+\beta A$, where $\alpha , \beta \in \mathbb {C}$.

Let

Show that every real matrix $B$ such that $AB = BA$ has the form

for some real numbers $a$, $b$, and $c$.

A square matrix $A$ is nilpotent if $A^{k} = 0$ for some positive integer $k$.

Consider the family of square matrices $A(\theta )$ defined by the solution of the matrix differential equation

with the initial condition $A(0) = I$, where $B$ is a constant square matrix.

Let $A$ be an $r \times r$ matrix of real numbers. Prove that the infinite sum

of matrices converges (i.e., for each $i, j$, the sum of $(i,j)$th entries converges), and hence that $e^{A}$ is a well-defined matrix.

Show that

for any complex $n \times n$ matrix $M$, where $\exp (M)$ is defined as in Problem 7.9.23.

Let $T$ be an $n \times n$ complex matrix. Show that

if and only if all the eigenvalues of $T$ have absolute value less than $1$.

Let $A$ and $B$ be $n \times n$ complex matrices. Prove that

Let $A$ and $B$ be $n \times n$ real matrices, and $k$ a positive integer. Find

Let $A$ be an $n \times n$ complex matrix such that $\operatorname {tr} A^{k} = 0$ for $k = 1, \ldots , n$. Prove that $A$ is nilpotent.

Let $N$ be a nilpotent complex matrix. Let $r$ be a positive integer. Show that there is a $n \times n$ complex matrix $A$ with

Let $A = (a_{ij})_{i,j=1}^{n}$ be a real $n \times n$ matrix such that $a_{ii} \geq 1$ for all $i$, and

Prove that $A$ is invertible.

Show that an $n \times n$ matrix of complex numbers $A = (a_{ij})$ satisfying

for $1 \leq i \leq n$ must be invertible.

Let $A = (a_{ij})_{i,j=1}^{n}$ be an $n \times n$ matrix such that $\sum_{j=1}^{n} \left|a_{ij}\right| < 1$ for each $i$. Prove that $I-A$ is invertible.

Let $A$ be a real $n \times n$ matrix. Let $M$ denote the maximum of the absolute values of the eigenvalues of $A$.

Let $A$ be an $n \times n$ matrix over $\mathbb {C}$ whose minimal polynomial has degree $k$.

Let $\left\| x\right\|$ denote the Euclidean norm of a vector $x$. Show that for any real $m \times n$ matrix $M$ there is a unique non-negative scalar $\sigma$, and (possibly non-unique) unit vectors $u \in \mathbb {R}^{m}$ and $v \in \mathbb {R}^{n}$ such that