Symmetric and Skew Symmetric Matrices

Matrix knowledge is required in a variety of fields of mathematics. If the transpose of a matrix equals the original matrix, the matrix is said to be a Skew-Symmetric Matrix. The resulting matrix is known as the transpose of the original matrix when the locations of the elements in the rows and columns are swapped, and the row becomes a column and vice versa. A single inverted comma is used to indicate it. A rectangular matrix’s transpose can never be equal to the original matrix. 

For example, if we have a matrix of order 2 x 3, the transpose will be of order 3 x 2, resulting in a new matrix. As a result, a rectangular matrix is never symmetrical. The transpose matrix for every square matrix, A, is represented as AT. As a result, a skew-symmetric or antisymmetric matrix A can be written as A = -AT. The use of a skew-symmetric matrix may be found in various domains, including machine learning and statistical analysis.

Skew-Symmetric Matrix

A skew-symmetric matrix is a type of square matrix that generally has no real-world applications but can be found in various mathematical and social sciences. To understand this, it’s important to understand how to determine the transpose of a matrix. The transpose of a matrix is what the name suggests: a transformation that flips an object over to its opposite-positioned side. 

A skew-symmetric matrix is a square matrix that equals the transpose matrix’s negative. To properly grasp a skew-symmetric matrix, it is necessary to learn how to find the transpose of a matrix. We’ve looked at matrix A in this case. 

How to Check Whether a Matrix is Symmetric?

Step 1: Determine the matrix’s transpose.

Step 2: Verify that the matrix’s transpose is the same as the original matrix.

Step 3: The matrix is symmetric if the transpose and original matrices are equivalent.

How to Check Whether a Matrix is Skew Symmetric?

Step 1: Determine the transpose of the matrix that was initially provided.

Step 2 – Next, discover the original matrix’s negative.

Step 3 – The matrix is said to be skew-symmetric if the negative of the matrix derived in Step 2 equals the transpose of the matrix.

Properties and Skew-Symmetric Matrix Example

The fundamental requirements for a skew-symmetric matrix are that it should be a square matrix, that is, the number of columns and rows must be equal, and that the provided matrix equals the negative of its transpose. A skew-symmetric matrix has a number of essential characteristics.

When two skew-symmetric matrices are combined, the resulting matrix is always skew-symmetric. If we take two skew-symmetric matrices A and B, where AT = -Aand BT =-B, we get (A + B)T =-(A + B)

A skew-symmetric matrix’s trace equals zero, which means that the total of all the components in the significant diagonal is likewise zero. 

A real skew-symmetric matrix A has a real eigenvalue of zero. A skew-symmetric matrix’s nonzero eigenvalues are hence non-real. 

Symmetric matrices, which could be inter-relation and covariance matrices, are commonly used in multivariate analysis.

A few symmetry-related characteristics in matrices are worth mentioning:

1. The product of any (not necessarily symmetric) matrix and its transpose is symmetric; that is, both AA′ and A′A are symmetric matrices.

2. If A is any square (not necessarily symmetric) matrix, then A + A′ is symmetric.

3. If A is symmetric and k is a scalar, then kA is a symmetric matrix.

5. The addition of any symmetric matrices number is known as symmetric.
6. The outcome of two symmetric matrices might not always be symmetric.

Theorems Related to Skew-Symmetric Matrices

• Theorem: For any square matrix Q with real number elements, Q + QTis a symmetric matrix, and Q – QTis a skew-symmetric matrix.

Proof:

Let P = Q + QT.

Transpose of P can be given as, PT = ( Q + QT)T = QT + ( QT )T = QT + Q = Q+ QT = P

⇒ Q+ QT is a symmetric matrix.

Next, we let R =  Q – QT

RT= ( Q  + ( –QT ))T= QT + ( –QT )T = QT – ( QT )T= QT – Q = -( Q – QT ) = -R

⇒ Q – QT is a skew-symmetric matrix.

Trace of Skew-Symmetric Matrices

A matrix’s trace equals the sum of all of its diagonal members. The diagonal elements of a skew-symmetric matrix are zero, as stated under the characteristics of a skew-symmetric matrix. As a result, a skew-symmetric matrix’s trace is always equal to zero. A square primary matrix is required for a skew-symmetric matrix, just as it is for a symmetric matrix. If the transpose of any matrix generates another matrix that is negative of the original matrix, it is said to be skew-symmetric. K’ = -K or k ab = -k ba is a skew-symmetric matrix in mathematics. In a skew-symmetric matrix, the entries in the diagonal are always equal to zero.

Conclusion

Among the most significant objects in linear algebra is the matrix. A matrix can represent a transformation, an operator, or a linear equation in some unknowns. The addition and scaling of the matrix, often known as a matrix operation, are two basic operations that may be done on matrices in mathematics.