All About Skew Symmetric Matrix

All About Skew Symmetric Matrix

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Vikram
Vikram Singh
Assistant Manager - Content
Updated on Oct 25, 2023 10:02 IST

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.

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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 = AT + BT
  • (kA)T = kAT, where k is some scalar
  • (AT)-1 = (A-1)T, where A is an invertible matrix
  • (AT)T = A
  • (AB)T = BTAT

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 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 = -AT

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Representation

If B = [bij]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.
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Example:

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

AT = -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 vTAv = 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 – AT will be a skew-symmetric matrix.

 Given: A is a real square matrix.

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

Concepts to be used:

  • (A+B)T = AT + BT
  • (kA)T = kAT, where k is some scalar
  • (AT)T = A

Proof:

Let B be a skew-symmetric matrix such that

B = A – AT

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.

Theorem – 2: Every Square matrix can be decomposed uniquely as a sum of the symmetric and skew-symmetric matrices.

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 BTAB 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) = kAT, where k is some scalar
  • (AT)T = A
  • (AB)T = BTAT

Proof:

(BTAB) 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.

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Conclusion

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

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FAQs

What is Skew-Symmetric Matrix?

A square matrix that is equal to the negative of its transpose is called a skew-symmetric matrix

What are the different properties of Skew-Symmetric Matrix?

1. The Sum of two skew-symmetric matrices is always a skew-symmetric matrix. 2. The scalar multiplication of a skew-symmetric matrix is always skew-symmetric. 3. If A and B are skew-symmetric matrices, such that AB = -BA, then AB is a skew-symmetric matrix. 4. The diagonal elements of a skew-symmetric matrix is always zero. 5. The Sum of a skew-symmetric matrix and the identity matrix is always an invertible matrix.

How do you determine which matrix is a skew-symmetric matrix?

To check if a given matrix is a skew-symmetric matrix or not, you have to check two conditions only: 1. Given matrix must have been a square matrix. 2. Transpose of a matrix is equal to its negative times of the matrix, i.e., (A^T) = -A.

What is the eigen value of the skew-symmetric matrix?

The eigen value of skew symmetric matrix is either zero or purely imaginary.

Can a skew symmetric matrix be diagonalized?

Yes, a skew symmetric matrix can be diagonalized, particularly over the complex field. The process involves finding a basis consisting of eigenvectors and then transforming the matrix into a diagonal form using these eigenvectors. The diagonal elements will be the eigenvalues, which are either zero or purely imaginary for real skew symmetric matrices.

How is a skew symmetric matrix used in real-world applications?

Skew symmetric matrices are widely used in areas such as physics, particularly in the study of angular momentum, and in the representation of cross products in vector spaces. They also appear in mathematical areas like differential geometry and the theory of Lie groups and Lie algebras.

Are skew symmetric matrices invertible?

Skew symmetric matrices can be invertible, but not always. A skew symmetric matrix is invertible if and only if its determinant is non-zero. However, for even order skew symmetric matrices, the determinant is always zero, making them non-invertible. For odd order, it depends on the specific elements of the matrix.

About the Author
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Vikram Singh
Assistant Manager - Content

Vikram has a Postgraduate degree in Applied Mathematics, with a keen interest in Data Science and Machine Learning. He has experience of 2+ years in content creation in Mathematics, Statistics, Data Science, and Mac... Read Full Bio

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