3.2 Transpose and special matrices

Once matrix multiplication is available, the shape of a matrix starts to matter as much as its numerical entries. The transpose is the first tool in this section that lets us read a matrix by its geometry: rows become columns, and the main diagonal becomes the axis of reflection. That simple idea explains symmetry, skew-symmetry, commuting patterns, and block structure.

Transpose swaps rows and columns

Definition

Transpose

If $A = [a_{ij}]$ is an $m \times n$ matrix, then its transpose $A^T$ is the $n \times m$ matrix whose (j,i) entry is $a_{ij}$ .

Equivalently,

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

So every row of $A$ becomes a column of $A^T$ , and every column of $A$ becomes a row.

This is not a new operation that changes the numbers in a mysterious way. It is purely a re-reading of the same entries in the opposite orientation.

Worked example

Compute a transpose

Let

A = \begin{bmatrix} 1 & 4 & -2 \\ 0 & 3 & 5 \end{bmatrix}.

Then

A^T = \begin{bmatrix} 1 & 0 \\ 4 & 3 \\ -2 & 5 \end{bmatrix}.

The $2 \times 3$ matrix becomes a $3 \times 2$ matrix because rows and columns swap roles.

Two basic identities should be treated as part of the definition-level vocabulary of the subject:

(A^T)^T = A, \qquad (A + B)^T = A^T + B^T, \qquad (cA)^T = cA^T.

When products are defined, transpose reverses the order:

(AB)^T = B^T A^T.

That reversal is not a cosmetic detail. It reflects the fact that matrix multiplication is built from row-column pairings, and transposition switches the role of rows and columns.

Worked example

Transpose identities in action

Take

A = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}, \qquad B = \begin{bmatrix} 3 & 0 \\ 4 & 5 \end{bmatrix}.

Then

A + B = \begin{bmatrix} 4 & 2 \\ 4 & 6 \end{bmatrix}, \qquad (A + B)^T = \begin{bmatrix} 4 & 4 \\ 2 & 6 \end{bmatrix}.

On the other hand,

A^T + B^T = \begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix} + \begin{bmatrix} 3 & 4 \\ 0 & 5 \end{bmatrix} = \begin{bmatrix} 4 & 4 \\ 2 & 6 \end{bmatrix}.

The product rule is similar, but the order matters:

AB = \begin{bmatrix} 11 & 10 \\ 4 & 5 \end{bmatrix}, \qquad (AB)^T = \begin{bmatrix} 11 & 4 \\ 10 & 5 \end{bmatrix},

while

B^T A^T = \begin{bmatrix} 3 & 4 \\ 0 & 5 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix} = \begin{bmatrix} 11 & 4 \\ 10 & 5 \end{bmatrix}.

Theorem

Basic properties of transpose

For compatible matrices:

$(A^T)^T = A$
$(A + B)^T = A^T + B^T$
$(cA)^T = cA^T$
$(AC)^T = C^T A^T$

The proof of the product rule is worth reading carefully once. To compare $(AC)^T$ and $C^T A^T$ , you compare their corresponding entries. The transpose does not change the numbers involved in the summation; it only changes the direction in which those numbers are arranged.

Symmetric and skew-symmetric matrices

Definition

Symmetric and skew-symmetric matrices

Let $A$ be a square matrix.

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

Symmetry says the matrix matches its reflection across the main diagonal. Skew-symmetry says the reflected matrix is the negative of the original.

Worked example

Recognize symmetry

The matrix

\begin{bmatrix} 2 & -1 & 4 \\ -1 & 3 & 0 \\ 4 & 0 & 5 \end{bmatrix}

is symmetric, because the (i,j) and (j,i) entries agree.

The matrix

\begin{bmatrix} 0 & 2 & -1 \\ -2 & 0 & 4 \\ 1 & -4 & 0 \end{bmatrix}

is skew-symmetric, because the transpose changes every off-diagonal entry's sign and leaves the diagonal as 0.

Three quick consequences are especially useful:

every diagonal entry of a real skew-symmetric matrix is 0
the zero matrix is both symmetric and skew-symmetric
the identity matrix is symmetric

The third item is a special case of the fact that diagonal matrices are fixed by transpose.

Theorem

Useful transpose-based identities

For every square matrix $A$ ,

(A + A^T)^T = A + A^T, \qquad (A - A^T)^T = -(A - A^T).

So $A + A^T$ is symmetric and $A - A^T$ is skew-symmetric.

Proof

Why the identities hold

The previous result is the starting point for the most important structural fact in this section.

Theorem

Decomposition into symmetric and skew-symmetric parts

Every square matrix $A$ can be written uniquely as

A = S + K,

where $S$ is symmetric and $K$ is skew-symmetric. In fact,

S = \frac{1}{2}(A + A^T), \qquad K = \frac{1}{2}(A - A^T).

The proof is a good model for how the course turns rough intuition into a formal argument.

Worked example

Decompose a matrix

Let

A = \begin{bmatrix} 1 & 0 & 1 \\ 2 & -1 & 3 \\ 4 & 2 & 0 \end{bmatrix}.

Then

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

\frac{1}{2}(A + A^T) = \begin{bmatrix} 1 & 1 & \frac{5}{2} \\ 1 & -1 & \frac{5}{2} \\ \frac{5}{2} & \frac{5}{2} & 0 \end{bmatrix},

and

\frac{1}{2}(A - A^T) = \begin{bmatrix} 0 & -1 & -\frac{3}{2} \\ 1 & 0 & \frac{1}{2} \\ \frac{3}{2} & -\frac{1}{2} & 0 \end{bmatrix}.

The first matrix is symmetric, the second is skew-symmetric, and their sum is $A$ .

There is also a very useful symmetric pattern that appears whenever a product is paired with its transpose.

Theorem

Products with a transpose are symmetric

If the product is defined, then both $A^T A$ and $A A^T$ are symmetric.

Proof

One-line proof

This identity is one of the main reasons transpose shows up again in later topics such as orthogonality, projections, and least-squares style arguments.

Commuting and non-commuting matrices

Definition

Commuting matrices

Two matrices $A$ and $B$ of the same size commute if

AB = BA.

For addition, commutativity is automatic. For multiplication, it is exceptional. This is one of the most important differences between scalar algebra and matrix algebra.

The zero matrix and the identity matrix commute with every square matrix of the same size. Diagonal matrices of the same size also commute with one another, because their products remain diagonal and the diagonal entries multiply in the ordinary commutative way.

Worked example

A non-commuting pair

Take

A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}, \qquad B = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}.

Then

AB = \begin{bmatrix} 2 & 1 \\ 1 & 1 \end{bmatrix}, \qquad BA = \begin{bmatrix} 1 & 1 \\ 1 & 2 \end{bmatrix}.

So $AB \ne BA$ .

That kind of example is not a curiosity. It is the reason matrix identities must always preserve the order of factors.

Theorem

A useful skew-symmetric commuting test

If $A$ and $B$ are skew-symmetric square matrices, then

AB \text{ is symmetric } \iff AB = BA.

Proof

Why this is true

Worked example

Symmetric times skew-symmetric, with commutation

Let

S = \begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix}, \qquad K = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}.

Here $S$ is symmetric and $K$ is skew-symmetric, but they do not commute:

SK = \begin{bmatrix} 0 & 2 \\ -3 & 0 \end{bmatrix}, \qquad KS = \begin{bmatrix} 0 & 3 \\ -2 & 0 \end{bmatrix}.

If two matrices of this kind do commute, then their product is skew-symmetric, because

(SK)^T = K^T S^T = (-K)S = -KS = -SK.

Special matrices

Special matrices are defined by where the zero entries must be. That makes them much easier to read and often much easier to multiply.

Definition

Diagonal, upper-triangular, and lower-triangular matrices

A diagonal matrix has all off-diagonal entries equal to 0.
An upper-triangular matrix has all entries below the main diagonal equal to 0.
A lower-triangular matrix has all entries above the main diagonal equal to 0.

Worked example

Transpose of a triangular matrix

U = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{bmatrix},

then

U^T = \begin{bmatrix} 1 & 0 & 0 \\ 2 & 4 & 0 \\ 3 & 5 & 6 \end{bmatrix}.

So the transpose of an upper-triangular matrix is lower-triangular.

Two structural facts are worth keeping in mind:

the product of two diagonal matrices is diagonal
the product of two upper-triangular matrices is upper-triangular
the product of two lower-triangular matrices is lower-triangular

For diagonal matrices, the product is especially simple:

\operatorname{diag}(d_1,\dots,d_n)\, \operatorname{diag}(e_1,\dots,e_n) = \operatorname{diag}(d_1 e_1,\dots,d_n e_n).

That is why diagonal matrices of the same size commute with each other.

Common mistake

Do not confuse the matrix with its zero pattern

An upper-triangular matrix may have many nonzero entries above the diagonal, but never below it. A diagonal matrix is more restrictive: every off-diagonal entry must be zero.

Block matrices

Large matrices are often easier to read if we group entries into blocks. A block matrix is still just a matrix; the blocks are a way of organizing it.

Definition

Block matrix

If a matrix is partitioned by drawing horizontal and vertical cuts, the resulting submatrices are called blocks.

For example,

A = \begin{bmatrix} 1 & 0 & 2 & 1 \\ 3 & 4 & 0 & -1 \\ 5 & 2 & 6 & 3 \\ 7 & 8 & 1 & 0 \end{bmatrix} = \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix},

where

A_{11} = \begin{bmatrix} 1 & 0 \\ 3 & 4 \end{bmatrix}, \quad A_{12} = \begin{bmatrix} 2 & 1 \\ 0 & -1 \end{bmatrix}, \quad A_{21} = \begin{bmatrix} 5 & 2 \\ 7 & 8 \end{bmatrix}, \quad A_{22} = \begin{bmatrix} 6 & 3 \\ 1 & 0 \end{bmatrix}.

At the block level, transpose simply swaps the blocks and transposes each block:

A^T = \begin{bmatrix} A_{11}^T & A_{21}^T \\ A_{12}^T & A_{22}^T \end{bmatrix}.

This is the same row/column swap principle as before, just applied to grouped pieces instead of individual entries.

Worked example

Block multiplication

Suppose the partition sizes are compatible, and write

M = \begin{bmatrix} A & B \\ C & D \end{bmatrix}, \qquad N = \begin{bmatrix} E & F \\ G & H \end{bmatrix}.

Then

MN = \begin{bmatrix} AE + BG & AF + BH \\ CE + DG & CF + DH \end{bmatrix}.

The formula looks exactly like ordinary $2 \times 2$ matrix multiplication, but each entry is now a block product.

The key condition is compatibility: the inner block sizes must match so that each block product is meaningful. If the partitions do not align, block multiplication is not even defined.

Common mistakes

Common mistake

Transpose reverses products

The correct identity is $(AB)^T = B^T A^T$ , not $A^T B^T$ .

Common mistake

Symmetric does not mean commuting

Symmetric is a property of one matrix. Commuting is a property of two matrices. They are different statements.

Common mistake

A skew-symmetric real matrix has a zero diagonal

If $A^T = -A$ , then each diagonal entry must equal its own negative, so every diagonal entry is 0.

Common mistake

Block sizes must line up

You may only multiply block matrices when the blocks are partitioned compatibly. Treating blocks as letters is only valid when the block products actually make sense.

Quick checks

Quick check

If $A$ is $3 × 2$ , what is the size of $A^T$ ?

Swap rows and columns.

Solution

Answer

Quick check

What must every diagonal entry of a real skew-symmetric matrix be?

Use $A^T = -A$ on the diagonal.

Solution

Answer

Quick check

Which identity is correct: $(AB)^T = A^T B^T$ or $(AB)^T = B^T A^T$ ?

Remember that transpose reverses the order of multiplication.

Solution

Answer

Quick check

If $S$ is symmetric and $K$ is skew-symmetric, what is $S^T$ and what is $K^T$ ?

Read the definitions directly.

Solution

Answer

Guided exercises

Quick check

Find the symmetric and skew-symmetric parts of $A = [[2,1],[4,3]]$ .

Use $S = 1/2 (A + A^T)$ and $K = 1/2 (A - A^T)$ .

Solution

Guided solution

Quick check

Let $A$ and $B$ be symmetric matrices. If $AB = BA$ , why is `AB` symmetric?

Use the transpose identity and the commuting assumption.

Solution

Guided solution

This note builds on 3.1 Matrix multiplication and identity matrices. For solution-set structure, continue to 4.1 Homogeneous systems and null space.

3.2 Transpose and special matrices

MATH1030: Linear algebra I

Transpose swaps rows and columns

Transpose

Compute a transpose

Transpose identities in action

Basic properties of transpose

Symmetric and skew-symmetric matrices

Symmetric and skew-symmetric matrices

Recognize symmetry

Useful transpose-based identities

Why the identities hold

Decomposition into symmetric and skew-symmetric parts

Decompose a matrix

Products with a transpose are symmetric

One-line proof

Commuting and non-commuting matrices

Commuting matrices

A non-commuting pair

A useful skew-symmetric commuting test

Why this is true

Symmetric times skew-symmetric, with commutation

Special matrices

Diagonal, upper-triangular, and lower-triangular matrices

Transpose of a triangular matrix

Do not confuse the matrix with its zero pattern

Block matrices

Block matrix

Block multiplication

Common mistakes

Transpose reverses products

Symmetric does not mean commuting

A skew-symmetric real matrix has a zero diagonal

Block sizes must line up

Quick checks

If AAA is 3×23 × 23×2, what is the size of ATA^TAT?

Answer

What must every diagonal entry of a real skew-symmetric matrix be?

Answer

Which identity is correct: (AB)T=ATBT(AB)^T = A^T B^T(AB)T=ATBT or (AB)T=BTAT(AB)^T = B^T A^T(AB)T=BTAT?

Answer

If SSS is symmetric and KKK is skew-symmetric, what is STS^TST and what is KTK^TKT?

Answer

Guided exercises

Find the symmetric and skew-symmetric parts of A=[[2,1],[4,3]]A = [[2,1],[4,3]]A=[[2,1],[4,3]].

Guided solution

Let AAA and BBB be symmetric matrices. If AB=BAAB = BAAB=BA, why is AB symmetric?

Guided solution

Related notes

Section mastery checkpoint

Prerequisites

Key terms in this unit

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More notes in this series

If $A$ is $3 × 2$ , what is the size of $A^T$ ?

Which identity is correct: $(AB)^T = A^T B^T$ or $(AB)^T = B^T A^T$ ?

If $S$ is symmetric and $K$ is skew-symmetric, what is $S^T$ and what is $K^T$ ?

Find the symmetric and skew-symmetric parts of $A = [[2,1],[4,3]]$ .

Let $A$ and $B$ be symmetric matrices. If $AB = BA$ , why is `AB` symmetric?