Symmetric and Skew Symmetric Matrices: Overview, Questions, Preparation

What are symmetric and skew-symmetric matrices?

Symmetric Matrix: A symmetric matrix is a square matrix that remains the same when transposed. In other words, a matrix A is symmetric if A equals its transpose, denoted as A^T.

A square matrix A =  ${\left[a_{ij}\right]}_{n\times n}$ is called symmetric iff A’ = A i.e., iff

$a_{ji}$ =  $a_{ij}$ ,  $1\le i\le n$ ,  $1\le j\le n$ .

Skew-symmetric Matrix: A skew-symmetric matrix is a square matrix where its transpose is equal to its negative. In other words, if A is a skew-symmetric matrix, then A^T = -A. One important fact is that the diagonal elements of a skew-symmetric matrix are always zero. 

A square matrix A =  ${\left[a_{ij}\right]}_{n\times n}$ is called skew - symmetric iff A’ = – A i.e., iff

$a_{ji}$ = –  $a_{ij}$ ,  $1\le i\le n$ ,  $1\le j\le n$ .

For example,  $\left[\begin{array}{cc}\hfill 3\hfill & \hfill -1\hfill \\ \hfill -1\hfill & \hfill 5\hfill \end{array}\right]$ and  $\left[\begin{array}{ccc}\hfill a\hfill & \hfill h\hfill & \hfill g\hfill \\ \hfill h\hfill & \hfill b\hfill & \hfill f\hfill \\ \hfill g\hfill & \hfill f\hfill & \hfill c\hfill \end{array}\right]$ are symmetric matrices of orders 2x2 and 3x3, respectively.

 

Mathematically it can be proven that all the diagonal elements will be zero. To prove let’s assume i = j in the condition aij = -aji, resulting in aii = -aii, which implies aii = 0. 

For example,  $\left[\begin{array}{cc}\hfill 0\hfill & \hfill -5\hfill \\ \hfill 5\hfill & \hfill 0\hfill \end{array}\right]$ and  $\left[\begin{array}{ccc}\hfill 0\hfill & \hfill h\hfill & \hfill g\hfill \\ \hfill -h\hfill & \hfill 0\hfill & \hfill f\hfill \\ \hfill -g\hfill & \hfill -f\hfill & \hfill 0\hfill \end{array}\right]$ are skew - symmetric matrices of orders 2x2 and 3x3 respectively.

Properties of Symmetric Matrices

Symmetric matrices are always square matrices (order of n x n).

Symmetric matrix is always equal to its transpose (A = A^T) 

Diagonal elements don’t change in the transpose of the matrix. In other words, Symmetric matrix is always symmetrical respect to the main diagonal (top left to bottom right).

Properties of Skew Symmetric Matrices

Symmetric matrices are always square matrices (order of n x n).

A matrix A is skew-symmetric if its transpose, A^T, is equal to the negative of the original matrix ( A = -A^T) 

All diagonal elements of a skew-symmetric matrix are always zero. 

The off-diagonal elements (a_ijwhere i ≠ j) are negatives of each other (a_ij = -a_ji). 

Important Facts for Exams

A square matrix A is symmetric if A = A^T, where A^T is the transpose of A. This means that $a_{ij}$ = $a_{ji}$  for all i and j.

A square matrix A is skew-symmetric if A = - A^T. This means that $a_{ij}$ = $-a_{ji}$ for all i and j.

All the diagonal elements in a skew–symmetric matrix are zero.

The matrices A.A^T and A^T. A is symmetric matrix.

For any square matrix A, the matrix A + A^T is a symmetric matrix and A – A^T is a skew-symmetric matrix always.

NCERT Theorem 1: For any square matrix A with real number entries, A + A′ is a symmetric matrix and A – A′ is a skew symmetric matrix.

Proof: For any square matrix A with real number entries, A + A′ is a symmetric matrix

Let B = A + A′, then

B′ = (A + A′)′

= A′ + (A′)′ (as (A + B)′ = A′ + B′)

= A′ + A (as (A′)′ = A)

= A + A′ (as A + B = B + A) = B

Therefore, B = A + A′ is a symmetric matrix

Proof: For any square matrix A with real number entries, A - A′ is a symmetric matrix

Now, let C = A – A′

C′ = (A – A′)′ = A′ – (A′)′ (Why?)

= A′ – A (Why?) = – (A – A′) = – C

Therefore, C = A – A′ is a skew-symmetric matrix.

NCERT Theorem 2: Any square matrix can be expressed as the sum of a symmetric and a Skew-symmetric matrix. Proof:

Let A be a square matrix, then

A =  $\frac{1}{2}$ A +  $\frac{1}{2}$ A =  $\frac{1}{2}$ (A + A’) +  $\frac{1}{2}$ (A – A’) ----------(i)

$\Rightarrow$ A = P + Q (say)

Where P =  $\frac{1}{2}$ ( A + A’) and Q =  $\frac{1}{2}$ (A – A’)

We note that

P’ =  ${\left((\mathrm{}\mathrm{A}\mathrm{}+\mathrm{}\mathrm{A}\mathrm{’})\right)}^{\text{'}}$ =  $\frac{1}{2}$ ( A + A’)’ =  $\frac{1}{2}$ (A’ + (A’)’) =  $\frac{1}{2}$ (A’ + A) =  $\frac{1}{2}$ (A + A’) = P

And Q’ =  ${\left((\mathrm{}\mathrm{A}-\mathrm{}\mathrm{A}\mathrm{’})\right)}^{\text{'}}$ =  $\frac{1}{2}$ ( A – A’)’ =  $\frac{1}{2}$ (A’ – (A’)’) =  $\frac{1}{2}$ (A’ – A) = –  $\frac{1}{2}$ (A – A’) = –Q

$\Rightarrow$ P is symmetric matrix and Q is skew - skew-symmetric matrix.

Therefore, A is the sum of a symmetric matrix P and skew - skew-symmetric matrix Q.