19.11 - Symmetric and Skew Symmetric Matrices

In the previous section, we saw transpose of a matrix. In this section, we will see symmetric and skew symmetric matrices.

Symmetric Matrix

This can be written in 5 steps:
1. Consider any square matrix A. We know how to write it's transpose A'.
2. If A' is equal to A, then we say that, A is a symmetric matrix.
3. An example is given below:
Let A = $\left[\begin{array}{r}                        3    &{    \sqrt2    }    &{    11    }\\ \sqrt2    &{    5    }    &{    -8    }\\ 11    &{    -8    }    &{    9    }\\ \end{array}\right]$.
• If we write A', we will get A itself. So A is a symmetric matrix.
4. Consider any element say the $\sqrt2$ at second row, first column. Let us interchange the row and column. So we want first row, second column. What is the element at the interchanged position? It is also $\sqrt2$
5. This is applicable to any element of a symmetric matrix.
• We can write:
   ♦ Let a be any element of a symmetric matrix. Let it be at the row i, column j.
   ♦ The element at row j, column i will also be the same element a.

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Skew Symmetric Matrix

This can be written in 5 steps:
1. Consider any square matrix A. We know how to write it's transpose A'.
2. If A' is equal to −A, then we say that, A is a skew symmetric matrix.
3. An example is given below:
Let A = $\left[\begin{array}{r}                        0    &{    -3    }    &{    11    }\\ 3    &{    0    }    &{    -7    }\\ -11    &{    7    }    &{    0    }\\ \end{array}\right]$.
• If we write A', we will get:
A' = $\left[\begin{array}{r}                        0    &{    3    }    &{    -11    }\\ -3    &{    0    }    &{    7    }\\ 11    &{    -7    }    &{    0    }\\ \end{array}\right]$.

• We see that, A' = −A. So A is a skew symmetric matrix.
4. Consider any non-diagonal element in A. Say the 7 at third row, second column. Let us interchange the row and column. So we want second row, third column. What is the element at the interchanged position? It is -7.
5. This is applicable to any non-diagonal element of a skew symmetric matrix.
• We can write:
   ♦ Let a be any non-diagonal element of a skew symmetric matrix. Let it be at the row i, column j.
   ♦ The element at row j, column i will be the -ve of element a.
6.The property written in (5) is applicable to diagonal elements also.This can be explained in 3 steps:
(i) For any diagonal element, the row number and column number will be the same.
• So any diagonal element, can be represented as a_ii.
(ii) In a skew symmetric matrix, any a_ij must be equal to −a_ji.
• So any a_ii must be equal to −a_ii
• That means:
a_ii = −a_ii
⇒ 2a_ii = 0
⇒ a_ii = 0
(iii) We can write:
If A is a skew symmetric matrix, then all diagonal elements of A will be zero.

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Now we will see two theorems related to symmetric and skew symmetric matrices.

Theorem 1
For any square matrix A,
(i) (A + A') is a symmetric matrix
(ii) (A – A') is a skew symmetric matrix.

Proof for part (i):
$\begin{array}{ll} {~\color{magenta}    1    }    &{\text{Let}}    &{B}    & {~=~}    &{A+A'}    \\ {~\color{magenta}    2    }    &{\implies}    &{B'}    & {~=~}    &{(A+A')'}    \\ {~\color{magenta}    3    }    &{{}}    &{{}}    & {~=~}    &{A' + (A')'}    \\ {~\color{magenta}    4    }    &{{}}    &{{}}    & {~=~}    &{A' + A}    \\ {~\color{magenta}    5    }    &{{}}    &{{}}    & {~=~}    &{A+A'}    \\ {~\color{magenta}    6    }    &{{}}    &{{}}    & {~=~}    &{B}    \\ \end{array}$

◼ Remarks:
(3) (A+B)' = A' + B'
(4) (A')' = A
(5) A+B = B+A

• We see that, B' = B. That means, B is a symmetric matrix. That means, (A+A') is a symmetric matrix.

Proof for part (ii):
$\begin{array}{ll} {~\color{magenta}    1    }    &{\text{Let}}    &{C}    & {~=~}    &{A-A'}    \\ {~\color{magenta}    2    }    &{\implies}    &{C'}    & {~=~}    &{(A-A')'}    \\ {~\color{magenta}    3    }    &{{}}    &{{}}    & {~=~}    &{A' - (A')'}    \\ {~\color{magenta}    4    }    &{{}}    &{{}}    & {~=~}    &{A' - A}    \\ {~\color{magenta}    5    }    &{{}}    &{{}}    & {~=~}    &{-(A-A')}    \\ {~\color{magenta}    6    }    &{{}}    &{{}}    & {~=~}    &{-C}    \\ \end{array}$

◼ Remarks:
(3) (A−B)' = [A+(−B)]'= [A+(−1B)]'= [A'+(−1B)']
= [A'+(−1)(B)']  =   A' − B'
(4) (A')' = A
(5) A−B = [A + (−B)] = [(−B) + A] = −[B − A]

• We see that, C' = −C. That means, C is a skew symmetric matrix. That means, (A−A') is a skew symmetric matrix.

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In the next section, we will see the second theorem.

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