Some More Special Matrices

DEFINITION 1.3.1  

1. A matrix $A$ over ${\mathbb{R}}$ is called symmetric if $A^{t} = A$ and skew-symmetric if $A^{t} = -A.$
2. A matrix $A$ is said to be orthogonal if $A A^t = A^t A = I.$

EXAMPLE 1.3.2  

1. Let $A = \begin{bmatrix}1 & 2 & 3 \\ 2 & 4 & -1 \\ 3 & -1 & 4 \end{bmatrix}$ and $B = \begin{bmatrix}0 & 1 & 2 \\ -1 & 0 & -3 \\ -2 & 3 & 0 \end{bmatrix}.$ Then $A$ is a symmetric matrix and $B$ is a skew-symmetric matrix.
2. Let $A = \begin{bmatrix}\frac{1}{\sqrt{3}} &\frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}}... ...\frac{1}{\sqrt{6}} & \frac{1}{\sqrt{6}} & - \frac{2}{ \sqrt{6} } \end{bmatrix}.$ Then $A$ is an orthogonal matrix.
3. Let $A=[a_{ij}]$ be an $n \times n$ matrix with $a_{ij} = \begin{cases}1 & {\mbox{ if }} i= j+1 \\ 0 &{\mbox{ otherwise }} \end{cases}.$ Then $A^n = {\mathbf 0}$ and $A^{\ell} \neq {\mathbf 0}$ for $1 \leq \ell \leq n-1.$ The matrices $A$ for which a positive integer $k$ exists such that $A^k = {\mathbf 0}$ are called NILPOTENT matrices. The least positive integer $k$ for which $A^k = {\mathbf 0}$ is called the ORDER OF NILPOTENCY.
4. Let $A = \begin{bmatrix}1 & 0 \\ 0 & 0 \end{bmatrix}.$ Then $A^2 = A.$ The matrices that satisfy the condition that $A^2 = A$ are called IDEMPOTENT matrices.

EXERCISE 1.3.3  

1. Show that for any square matrix $A,$ $S = \frac{1}{2} (A+A^{t})$ is symmetric, $T = \frac{1}{2} (A - A^{t})$ is skew-symmetric, and $A = S + T.$
2. Show that the product of two lower triangular matrices is a lower triangular matrix. A similar statement holds for upper triangular matrices.
3. Let $A$ and $B$ be symmetric matrices. Show that $A B$ is symmetric if and only if $A B = B A.$
4. Show that the diagonal entries of a skew-symmetric matrix are zero.
5. Let $A, B$ be skew-symmetric matrices with $A B = B A.$ Is the matrix $A B$ symmetric or skew-symmetric?
6. Let $A$ be a symmetric matrix of order $n$ with $A^{2} = {\mathbf 0}.$ Is it necessarily true that $A = {\mathbf 0}?$
7. Let $A$ be a nilpotent matrix. Show that there exists a matrix $B$ such that $B(I + A) = I = (I+A)B.$