Transpose and symmetric matrices

The transpose of a matrix is obtained by interchanging its rows and columns. Although the operation is simple, it interacts beautifully with addition and multiplication, and it singles out two structurally important families: symmetric matrices, where $A^T = A$, and skew-symmetric matrices, where $A^T = -A$. Every square matrix can be uniquely decomposed into a symmetric and a skew-symmetric part , a fact you will use throughout the chapter.

Definition

If $A = (a_{ij})_{m \times n}$, its transpose $A^T$ (also written $A'$) is the $n \times m$ matrix

$(A^T)_{ij} = a_{ji}.$

So row $i$ of $A$ becomes column $i$ of $A^T$.

Example

$A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix} \Rightarrow A^T = \begin{pmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{pmatrix}$.

Algebraic properties of transpose

1. $(A^T)^T = A$.
2. $(A + B)^T = A^T + B^T$.
3. $(\lambda A)^T = \lambda A^T$.
4. $(AB)^T = B^T A^T$.

The reversal in (4) is the most-tested fact. It mirrors the inverse formula $(AB)^{-1} = B^{-1} A^{-1}$.

Proof of (4). Both sides are $n \times m$ matrices (if $A$ is $m \times p$ and $B$ is $p \times n$). The $(i, j)$ entry of $(AB)^T$ is the $(j, i)$ entry of $AB$, which is $\sum_k a_{jk} b_{ki}$. The $(i, j)$ entry of $B^T A^T$ is $\sum_k (B^T)_{ik}(A^T)_{kj} = \sum_k b_{ki} a_{jk}$. Same expression. $\blacksquare$

Symmetric matrices

A square matrix $A$ is symmetric if $A^T = A$, i.e., $a_{ij} = a_{ji}$ for all $i, j$. Symmetric matrices are precisely those that look the same when reflected across the main diagonal.

Examples: any diagonal matrix, $\begin{pmatrix} 1 & 2 \\ 2 & 3 \end{pmatrix}$, the identity.

The set of $n \times n$ symmetric matrices is closed under addition and scalar multiplication, but not under multiplication: $A, B$ symmetric does not imply $AB$ symmetric.

Skew-symmetric matrices

A square matrix $A$ is skew-symmetric if $A^T = -A$, i.e., $a_{ij} = -a_{ji}$. Setting $i = j$ shows $a_{ii} = 0$: all diagonal entries are zero.

Examples: $\begin{pmatrix} 0 & 2 \\ -2 & 0 \end{pmatrix}$, $\begin{pmatrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{pmatrix}$.

The decomposition theorem

Theorem. Every square matrix $A$ can be written uniquely as $A = P + Q$, where $P$ is symmetric and $Q$ is skew-symmetric.

Proof. Set $P = \tfrac{1}{2}(A + A^T)$ and $Q = \tfrac{1}{2}(A - A^T)$. Then $A = P + Q$. Check $P^T = \tfrac{1}{2}(A^T + A) = P$, so $P$ symmetric. Check $Q^T = \tfrac{1}{2}(A^T - A) = -Q$, so $Q$ skew-symmetric.

Uniqueness: suppose $A = P_1 + Q_1 = P_2 + Q_2$. Then $P_1 - P_2 = Q_2 - Q_1$. The left side is symmetric, the right side skew-symmetric. The only matrix that is both is the zero matrix. So $P_1 = P_2, Q_1 = Q_2$. $\blacksquare$

Worked decomposition

Let $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$. Then $A^T = \begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix}$.

$P = \tfrac{1}{2}(A + A^T) = \tfrac{1}{2}\begin{pmatrix} 2 & 5 \\ 5 & 8 \end{pmatrix} = \begin{pmatrix} 1 & 5/2 \\ 5/2 & 4 \end{pmatrix}$. Symmetric.

$Q = \tfrac{1}{2}(A - A^T) = \tfrac{1}{2}\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 0 & -1/2 \\ 1/2 & 0 \end{pmatrix}$. Skew-symmetric.

Verify $P + Q = A$. $\checkmark$

Worked examples

Example 1. Find $A^T$ for $A = \begin{pmatrix} 2 & -1 & 3 \\ 4 & 0 & -2 \end{pmatrix}$.

$A^T = \begin{pmatrix} 2 & 4 \\ -1 & 0 \\ 3 & -2 \end{pmatrix}$.

Example 2. Verify $(AB)^T = B^T A^T$ for $A = \begin{pmatrix} 1 & 2 \end{pmatrix}, B = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$.

$AB = (3 + 8) = (11)$, so $(AB)^T = (11)$. $B^T = (3, 4)$, $A^T = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$, so $B^T A^T = 3 + 8 = (11)$. $\checkmark$

Example 3. Show that $A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 5 & 4 \\ 3 & 4 & 7 \end{pmatrix}$ is symmetric.

Compute $A^T$: $\begin{pmatrix} 1 & 2 & 3 \\ 2 & 5 & 4 \\ 3 & 4 & 7 \end{pmatrix}$, same as $A$.

Example 4. Decompose $\begin{pmatrix} 2 & 3 \\ 1 & 0 \end{pmatrix}$ into symmetric and skew-symmetric parts.

$A^T = \begin{pmatrix} 2 & 1 \\ 3 & 0 \end{pmatrix}$. $P = \tfrac{1}{2}\begin{pmatrix} 4 & 4 \\ 4 & 0 \end{pmatrix} = \begin{pmatrix} 2 & 2 \\ 2 & 0 \end{pmatrix}$. $Q = \tfrac{1}{2}\begin{pmatrix} 0 & 2 \\ -2 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$. Verify $P + Q = A$.

Example 5. Show: if $A$ is square then $A + A^T$ is symmetric and $A - A^T$ is skew-symmetric.

$(A + A^T)^T = A^T + A = A + A^T$. $(A - A^T)^T = A^T - A = -(A - A^T)$. $\checkmark$

Example 6. Show that the product of two symmetric matrices is symmetric iff they commute.

$(AB)^T = B^T A^T = BA$. So $AB$ is symmetric iff $AB = BA$.

Try it yourself

1. Write the transpose of $\begin{pmatrix} 1 & 0 & -1 \\ 2 & 3 & 4 \end{pmatrix}$.
2. Determine whether $\begin{pmatrix} 0 & 1 & 2 \\ -1 & 0 & 3 \\ -2 & -3 & 0 \end{pmatrix}$ is symmetric or skew-symmetric.
3. Show: $A$ is symmetric iff $A + A^T = 2A$.
4. Show: $A$ is skew-symmetric iff $A + A^T = O$.
5. Decompose $\begin{pmatrix} 1 & 5 \\ -1 & 2 \end{pmatrix}$ into symmetric and skew-symmetric parts.
6. Verify $(AB)^T = B^T A^T$ for $A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & 3 \\ 4 & 5 \end{pmatrix}$.
7. Show: if $A$ is symmetric and invertible, then $A^{-1}$ is symmetric.
8. Find $a$ if $\begin{pmatrix} 0 & a \\ -2 & 0 \end{pmatrix}$ is to be skew-symmetric.
9. Show: every diagonal entry of a skew-symmetric matrix is zero.
10. If $A$ is skew-symmetric of order $n$ and $n$ is odd, what is $\det A$? (Preview from Chapter 4.)
11. Find $A^T A$ for $A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ 1 & 1 \end{pmatrix}$.
12. Show that $A^T A$ is always symmetric, for any matrix $A$.
13. Decompose $\begin{pmatrix} 3 & -2 & -4 \\ 3 & -2 & -5 \\ -1 & 1 & 2 \end{pmatrix}$ into symmetric and skew-symmetric parts.
14. Prove: if $A$ and $B$ are both skew-symmetric, then so is $\alpha A + \beta B$ for any scalars.

Pitfalls / Tricks

• $(AB)^T = B^T A^T$ , the order reverses. Forgetting this is a classical error.
• A skew-symmetric matrix has zero diagonal; a symmetric matrix can have anything on the diagonal.
• The decomposition $A = P + Q$ is unique. The $1/2$ factor is essential.
• Product of two symmetric matrices is symmetric only if they commute.
• $A^T A$ and $AA^T$ are always symmetric, but they are different matrices in general (and need not be equal even in order).

Next, elementary row operations and inverses.