Special matrices: projection, orthogonal, idempotent and partitioned

A map of the special matrices

Type	Defining property	Eigenvalues	Determinant
Symmetric	$A^\top = A$	real	product of eigenvalues
Skew-symmetric	$A^\top = -A$	0 or purely imaginary	0 when the order is odd
Orthogonal	$Q^\top Q = I$	$\lvert\lambda\rvert = 1$	$\pm 1$
Idempotent	$P^2 = P$	0 or 1	0 or 1
Orthogonal projection	$P^2 = P = P^\top$	0 or 1	0 unless $P = I$
Nilpotent	$N^k = 0$ for some $k$	all 0	0
Involutory	$A^2 = I$	$\pm 1$	$\pm 1$
Triangular	zeros below (or above) the diagonal	the diagonal entries	product of the diagonal
Diagonal	zeros off the diagonal	the diagonal entries	product of the diagonal

How the eigenvalue rules are proved. If $Av = \lambda v$ with $v \ne 0$, then any polynomial identity in $A$ also holds for $\lambda$. For example:

• $P^2 = P$ gives $\lambda^2 v = \lambda v$, so $\lambda^2 = \lambda$ and $\lambda \in \{0, 1\}$;
• $N^k = 0$ gives $\lambda^k = 0$;
• $A^2 = I$ gives $\lambda^2 = 1$.

Symmetric and skew-symmetric matrices

Every square matrix splits uniquely into a symmetric part and a skew-symmetric part:

$A = \underbrace{\tfrac12(A + A^\top)}_{\text{symmetric}} + \underbrace{\tfrac12(A - A^\top)}_{\text{skew}} .$

Example 1. $A = \begin{pmatrix} 1 & 4 \\ 2 & 3 \end{pmatrix}$ has symmetric part $\begin{pmatrix} 1 & 3 \\ 3 & 3 \end{pmatrix}$ and skew part $\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$.

Spectral theorem. A real symmetric matrix has real eigenvalues, and it has an orthonormal basis of eigenvectors. So

$A = Q \Lambda Q^\top = \sum_i \lambda_i\, q_i q_i^\top ,$

with $Q$ orthogonal and $\Lambda$ diagonal. Eigenvectors belonging to different eigenvalues are automatically orthogonal.

Why. If $Av = \lambda v$ and $Aw = \mu w$ with $\lambda \ne \mu$, then $\lambda\, v^\top w = (Av)^\top w = v^\top A w = \mu\, v^\top w$, so $v^\top w = 0$.

Skew-symmetric facts.

• The diagonal is zero, since $a_{ii} = -a_{ii}$.
• $x^\top A x = 0$ for every $x$.
• For odd $n$, $\det A = \det A^\top = \det(-A) = (-1)^n \det A = -\det A$, so $\det A = 0$.

Example 2. $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ has $\lambda^2 + 1 = 0$, so $\lambda = \pm i$, which is purely imaginary. Any $3 \times 3$ skew-symmetric matrix has determinant 0.

Products.

• $A^\top A$ and $AA^\top$ are always symmetric, with eigenvalues $\ge 0$.
• The product of two symmetric matrices is symmetric only if they commute: $(AB)^\top = BA$.

Orthogonal matrices

Definition. $Q$ is orthogonal if $Q^\top Q = I$, so $Q^{-1} = Q^\top$. Its columns are orthonormal, and so are its rows.

Properties.

• Lengths and angles are kept: $\lVert Qx\rVert = \lVert x\rVert$ and $(Qx)\cdot(Qy) = x \cdot y$.
• $\det Q = \pm 1$. $+1$ means a rotation and $-1$ means a reflection is involved.
• Every eigenvalue has $\lvert\lambda\rvert = 1$.
• Products and inverses of orthogonal matrices are orthogonal.
• Permutation matrices are orthogonal.

Examples.

• Rotation by $\theta$: $R = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$, with $\det R = 1$ and eigenvalues $e^{\pm i\theta}$.
• Reflection across the line $y = x$: $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$, with determinant $-1$ and eigenvalues $1$ (along $(1, 1)$) and $-1$ (along $(1, -1)$).
• Householder reflection: $H = I - 2\dfrac{uu^\top}{u^\top u}$. It reflects across the hyperplane perpendicular to $u$, and it is symmetric, orthogonal and involutory.

Example 3. With $u = (1, 1)$: $H = I - \begin{pmatrix} 1 & 1 \\ 1 & 1\end{pmatrix} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$.

• $H(1, 1) = (-1, -1)$: the vector $u$ is flipped.
• $H(1, -1) = (1, -1)$: vectors perpendicular to $u$ stay put.