Special Matrices

Overview

Certain matrices have special structures that give them useful properties. Recognizing these special forms simplifies many calculations.

Symmetric Matrices

A matrix $A$ is symmetric if $A^T = A$.

$A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 5 & 6 \\ 3 & 6 & 9 \end{bmatrix}$

Properties

• Only square matrices can be symmetric
• $a_{ij} = a_{ji}$ for all $i, j$
• Eigenvalues are real
• Eigenvectors for distinct eigenvalues are orthogonal
• $A + A^T$ is always symmetric

Skew-Symmetric Matrices

A matrix $A$ is skew-symmetric if $A^T = -A$.

$A = \begin{bmatrix} 0 & 2 & -3 \\ -2 & 0 & 4 \\ 3 & -4 & 0 \end{bmatrix}$

Properties

• Diagonal entries are zero
• $a_{ij} = -a_{ji}$
• $A - A^T$ is always skew-symmetric
• Any matrix $A = \frac{A + A^T}{2} + \frac{A - A^T}{2}$ (symmetric + skew-symmetric)

Orthogonal Matrices

A matrix $Q$ is orthogonal if $Q^T Q = QQ^T = I$.

Equivalently: $Q^T = Q^{-1}$

Properties

• Columns are orthonormal vectors
• Rows are orthonormal vectors
• $\lvert\det(Q)\rvert = 1$
• Preserves lengths: $\|Q\mathbf{x}\| = \|\mathbf{x}\|$
• Preserves angles

Examples

Rotation matrix:

$Q = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$

Triangular Matrices

Upper Triangular

All entries below main diagonal are zero:

$U = \begin{bmatrix} u_{11} & u_{12} & u_{13} \\ 0 & u_{22} & u_{23} \\ 0 & 0 & u_{33} \end{bmatrix}$

Lower Triangular

All entries above main diagonal are zero:

$L = \begin{bmatrix} l_{11} & 0 & 0 \\ l_{21} & l_{22} & 0 \\ l_{31} & l_{32} & l_{33} \end{bmatrix}$

Properties

• Product of upper triangular matrices is upper triangular
• Product of lower triangular matrices is lower triangular
• Determinant is product of diagonal entries
• Eigenvalues are the diagonal entries

Diagonal Matrices

Only diagonal entries are non-zero:

$D = \begin{bmatrix} d_1 & 0 & 0 \\ 0 & d_2 & 0 \\ 0 & 0 & d_3 \end{bmatrix}$

Properties

• Easy to compute powers: $D^n$ has entries $d_i^n$
• Easy to invert: $D^{-1}$ has entries $1/d_i$ (if all $d_i \neq 0$)
• Diagonal matrices commute: $D_1 D_2 = D_2 D_1$

Nilpotent Matrices

A matrix $N$ is nilpotent if $N^k = O$ for some positive integer $k$.

$N = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}$

$N^2 = \begin{bmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \qquad N^3 = O$

Idempotent Matrices

A matrix $P$ is idempotent if $P^2 = P$.

$P = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$

$P^2 = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} = P$

Eigenvalues are 0 or 1.

Involutory Matrices

A matrix $A$ is involutory if $A^2 = I$.

Equivalently: $A = A^{-1}$

$A = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$

$A^2 = I$

Summary Table

Type	Definition	Key Property
Symmetric	$A^T = A$	Real eigenvalues
Skew-symmetric	$A^T = -A$	Diagonal is zero
Orthogonal	$Q^T Q = I$	Preserves lengths
Upper triangular	$a_{ij} = 0$ for $i > j$	Easy back-substitution
Lower triangular	$a_{ij} = 0$ for $i < j$	Easy forward-substitution
Diagonal	$a_{ij} = 0$ for $i \neq j$	Easy powers/inverse
Nilpotent	$A^k = O$	Eigenvalues all zero
Idempotent	$A^2 = A$	Eigenvalues 0 or 1
Involutory	$A^2 = I$	$A = A^{-1}$