All About Skew Symmetric Matrix

A square matrix is said to be a skew-symmetric matrix if it is equal to the negative of its transpose. This article, we will briefly discuss skew-symmetric matrix, properties and theorems related to it.

A matrix is a rectangular arrangement of data points in rows and columns, and matrix with the same number of rows and columns is said to be a square matrix. This article will discuss a square matrix, which is equal to the negative of its transpose, i.e., a skew-symmetric matrix.

In the previous article, we discussed the symmetric matrix; the properties and theorems of the skew-symmetric matrix will be very similar to the symmetric matrix.

Before starting the article, let’s discuss some of the properties of the transpose of a matrix.

• (A+B)^T = A^T + B^T
• (kA)^T = kA^T, where k is some scalar
• (A^T)^-1 = (A^-1)^T, where A is an invertible matrix
• (A^T)^T = A
• (AB)^T = B^TA^T

We will use these properties in the chapter while proving the theorems.

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Now, let’s explore Skew-Symmetric Matrix, its definition, properties, and theorems.

Table of Content

• What is Skew-Symmetric Matrix
• Properties
• Theorems

What is a Skew-Symmetric Matrix?

Definition

A square matrix that is equal to the negative of its transpose is called a skew-symmetric matrix, i.e., a square matrix is said to be a skew-symmetric matrix if and only if:

A = -A^T

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Representation

If B = [b_ij]_n, is a skew-symmetric matrix, if bij = -bji, for all

• 1<= i, j <= n, for all n belongs to Natural Number.
• bij represents the element at the i th row and j th column.

Example:

Here, we have taken three examples: first two examples are skew-symmetric in nature, since

A^T = -A, and B ^T = -B.

But, in example – 3, C ^T != -C

Hence, C is not a skew-symmetric matrix.

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Properties

• The Sum of two skew-symmetric matrices is always a skew-symmetric matrix.

(A + B)^T = A ^T + B ^T = -A – B = – (A + B)

Hence, A + B = -(A + B) ^T

• The scalar multiplication of a skew-symmetric matrix is always skew-symmetric.

(kA)^T = k(A)^T = k(-A) = -kA

Hence, (kA)^T = -kA

• If A and B are skew-symmetric matrices, such that AB = -BA, then AB is a skew-symmetric matrix.

(AB) ^T = B ^T A ^T = (-B)(-A) = BA = -AB

Hence, (AB) ^T = -AB

• The diagonal elements of a skew-symmetric matrix is always zero.

Let A be a skew-symmetric matrix, then

aij = -aji

Now, for the diagonal entries, the above will be:

aii = -aii

aii + aii = 0

2aii = 0

aii = 0 

• The Trace (i.e., the sum of all elements of the principal diagonal) of a skew-symmetric matrix is always zero.
• The Sum of a skew-symmetric matrix and the identity matrix is always an invertible matrix.
• The inverse of an invertible skew-symmetric matrix is always a skew-symmetric matrix.

Let A be a skew-symmetric matrix:

A = -A ^T

Taking inverse both the side, we get:

A^-1 = (-A ^T ) ^-1 = -(A^-1) ^T

Hence, 

A^-1 = -(A^-1) ^T

• Let v be an n-dimensional column vector, then v^TAv = 0, where A is a skew-symmetric matrix.
• Determinant of a skew-symmetric matrix
• The eigenvalues of skew-symmetric are always zero or imaginary.

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Theorem

Theorem – 1: For any real square matrix A, A – A^T will be a skew-symmetric matrix.

 Given: A is a real square matrix.

To Prove: A – A^T is a skew-symmetric matrix.

Concepts to be used:

• (A+B)^T = A^T + B^T
• (kA)^T = kA^T, where k is some scalar
• (A^T)^T = A

Proof:

Let B be a skew-symmetric matrix such that

B = A – A^T

Now, taking the transpose both the side, we get

B ^T = (A – A ^T ) ^T = A ^T + (-A ^T ) ^T = A ^T – A = – (A – A ^T ) = -B

Hence, A – A^T is a skew-symmetric matrix.

i.e., for any square matrix A,

A = (½) [S +V], where 

S = A + A ^T , and V = A – A ^T

To know the proof of the above theorem, check out: All about Symmetric Matrix.

Theorem – 3: If A is a skew-symmetric matrix, then B^TAB is a skew-symmetric matrix.

Given: A is a skew-symmetric matrix, and B is any matrix.

To Prove: B ^T AB is a skew-symmetric

Concepts to be used:

• (kA) = kA^T, where k is some scalar
• (A^T)^T = A
• (AB)^T = B^TA^T

Proof:

(B^TAB) ^T = [ B ^T (AB)] ^T = (AB) ^T (B ^T ) ^T = B ^T A ^T B = B ^T (-A)B = -B ^T AB

Hence,

(B ^T AB) ^T = – -B ^T AB

Thus, if A is a skew-symmetric matrix, then B ^T AB is a skew-symmetric matrix.

Conclusion

In this article, we have briefly discussed skew-symmetric matrix, their properties and theorems related to it.

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