All About Skew Symmetric Matrix
A skew-symmetric matrix is a square matrix whose transpose is equal to its negative. In other words, it satisfies the condition AT = -A. This type of matrix has some interesting properties and is used in various fields, including physics, engineering, and computer science. Read on to learn more about skew-symmetric matrices and their applications.
A matrix is a rectangular arrangement of data points in rows and columns, and a 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 to prove the theorems.
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Now, let’s explore the Skew-Symmetric Matrix, its definition, properties, and theorems.
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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 of Skew-Symmetric Matrix
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: The 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 of Skew-Symmetric Matrix
- 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 principal diagonal elements) 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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Theorems on Skew-Symmetric Matrix
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 find 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.
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.
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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