Skew Symmetric Matrix
2026-02-28 13:21 Diff
https://brightchamps.com/en-us/math/algebra/skew-symmetric-matrix

In this section, we will be discussing the theorems on skew symmetric matrices. These theorems are useful in matrix decomposition, transformations, and applications. 

Theorem 1: For a square matrix A, A - AT is skew symmetric

For any square matrix A, we should prove that A - AT is skew symmetric.

We can take help from the properties of skew symmetric matrices, such as: 

  • (A + B)T = AT + BT
  • (kA)T = kAT
  • (AT)T = A
     

For A - AT  to be skew symmetric, let’s prove that B = A - AT
To prove B = A - AT, let’s take transpose on both sides.
BT = (A - AT)T = AT - A
BT = AT - (AT)T

We know that (AT)T = A 
Therefore, BT  = AT - A
BT  = -(A - AT)  BT = -B
So, -B = -(A - AT)

We can now say that B = (A - AT) is skew-symmetric.

Theorem 2: Decomposing a Square Matrix

Like all matrices, square matrices also encode complex relationships. It is important to decompose a square matrix into similar components to understand their properties and structure. A square matrix is expressed as the sum of a symmetric matrix, and a skew-symmetric matrix will be proved in this theorem. This decomposition is significant in applications like physics. 


In this theorem, we will be using the following properties:

  • (AT)T = A
  • (A + B)T = AT + BT
  • (kA)T = kAT
  • If MT = M, then matrix M is symmetric
  • If NT = -N, then matrix N is skew symmetric

If A is a square matrix, it can be written as:
A = \(\frac{1}{2}\) (A + AT) + 1/2 (A - AT)
Let’s consider, 
P = \(\frac{1}{2}\) (A + AT) 
Q = \(\frac{1}{2}\) (A - AT)

Taking the transpose of both P and Q
PT = (\(\frac{1}{2}\) (A + AT))T
= ½(AT + (AT)T)
= ½(AT + A)
P = ½(AT + A)
So, PT = P

QT = (\(\frac{1}{2}\) (A - AT))T
= ½(AT - (AT)T)
= ½ (AT - A)
= ½ (AT - A)
-Q = ½ (A - AT)
So, QT = - Q

So, the square matrix A can be written as the sum of a symmetric matrix P and a skew symmetric matrix Q. 

Theorem 3: For a skew symmetric matrix A and any matrix B, the matrix BTAB is skew symmetric.

Let’s take the help of the following properties:

  • (AB)T = BTAT
  • (AT)T = A
  •  AT = -A

In this theorem, we shall prove (BTAB)T = -BTAB
Start with:
(BT AB)T = (BT (AB))T

Now use the transpose of a product rule:
= (AB)T (BT)T

Since (AB)T = BT AT we get:
(BT AB)T = BTATB
 

Since AT = -A, we can make the substitution.
 

Therefore, (BT AB)T = BT(-A)B
(BT AB)T = -BT AB
 

So, if A is a skew symmetric matrix, then BT AB is a skew symmetric matrix.