Transpose, symmetric and skew-symmetric matrices

What you will be able to do

• write the transpose of a matrix and use its four rules;
• recognise symmetric and skew-symmetric matrices;
• write any square matrix as the sum of a symmetric and a skew-symmetric matrix.

The transpose

The transpose of a matrix $A$, written $A^T$ (some books write $A'$), is the matrix whose rows are the columns of $A$. If $A$ has order $m \times n$, then $A^T$ has order $n \times m$. The entry in row $i$ and column $j$ of $A^T$ is $a_{ji}$.

The rules of the transpose are:

• $(A^T)^T = A$;
• $(A + B)^T = A^T + B^T$;
• $(kA)^T = kA^T$;
• $(AB)^T = B^TA^T$, with the order reversed.

Check the last rule with $A =

$and$B =

$. We found$AB =

$, so$(AB)^T =

$. Also$B^TA^T =

=

$. They agree. The product$A^TB^T$is$

$ and does not.

Symmetric and skew-symmetric matrices

A square matrix $A$ is

• symmetric if $A^T = A$, which means $a_{ij} = a_{ji}$ for all $i, j$;
• skew-symmetric if $A^T = -A$, which means $a_{ij} = -a_{ji}$ for all $i, j$.

The picture of a symmetric matrix is a mirror along the main diagonal. For $i = j$ a skew-symmetric matrix gives $a_{ii} = -a_{ii}$, so $a_{ii} = 0$. Every diagonal entry of a skew-symmetric matrix is zero.

This matrix is symmetric. The next one is skew-symmetric.

Two more facts are used often. For any matrix $A$, the products $AA^T$ and $A^TA$ are symmetric, because $(AA^T)^T = (A^T)^TA^T = AA^T$. And if $A$ is symmetric, then so is every power $A^n$ and every multiple $kA$.

Symmetric plus skew-symmetric

For any square matrix $A$, put

Then $S$ is symmetric, because $S^T = \tfrac{1}{2}(A^T + A) = S$. And $K$ is skew-symmetric, because $K^T = \tfrac{1}{2}(A^T - A) = -K$. Adding gives $S + K = A$.

So every square matrix is the sum of a symmetric and a skew-symmetric matrix, and there is only one way to do it. If $A = S' + K'$ is another way, then $A^T = S' - K'$, and adding the two equations gives $S' = \tfrac{1}{2}(A + A^T) = S$.

A worked example

Write $A =

$ as the sum of a symmetric and a skew-symmetric matrix.

1. The transpose. $A^T =

$.

2. The symmetric part. $A + A^T =

$, so$S =

$.

3. The skew-symmetric part. $A - A^T =

$, so$K =

$.

4. Check. The top row of $S + K$ is $(3, 5, 1)$, which is the top row of $A$.

Practise with new numbers

Try it in Python

The program splits the matrix of the worked example into its two parts. It then checks that $S$ is symmetric, that $K$ is skew-symmetric and that they add up to $A$.

Common mistakes

• Not reversing the order. $(AB)^T = B^TA^T$, not $A^TB^T$.
• Using a non-zero diagonal in a skew-symmetric matrix. The diagonal must be all zeros.
• Calling a non-square matrix symmetric. Only a square matrix can have $A^T = A$.
• Forgetting the half. $S$ and $K$ both carry the factor $\tfrac{1}{2}$.
• Sign slips in $K$. $K$ uses $A - A^T$, so the entry in row $1$, column $2$ is $\tfrac{1}{2}(a_{12} - a_{21})$.

Remember these

• $(A^T)^T = A$, $(A + B)^T = A^T + B^T$, $(kA)^T = kA^T$ and $(AB)^T = B^TA^T$.
• Symmetric: $A^T = A$. Skew-symmetric: $A^T = -A$, with a zero diagonal.
• $A = \tfrac{1}{2}(A + A^T) + \tfrac{1}{2}(A - A^T)$, and this split is unique.
• $AA^T$ is always symmetric.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Write the transpose of $

$.

$.

2. Is $

$symmetric? Is$

$ skew-symmetric?

3. Why must every diagonal entry of a skew-symmetric matrix be zero?

4. If $A$ is symmetric, is $3A + A^T$ symmetric?

5. Find the symmetric and skew-symmetric parts of $A =

$.

$and$K =

$. Their sum is$A$.

References

• NCERT, Mathematics Textbook for Class XII, Part I, Chapter 3 (Matrices), the sections on the transpose and on symmetric and skew-symmetric matrices.
• OpenStax, Precalculus 2e, Section 9.5 (Matrices and Matrix Operations).