Symmetric and Skew Symmetric Matrices

In mathematics, a matrix is an array of numbers separated into rows and columns and arranged in a rectangular pattern. All the numbers are enclosed in square brackets. An antisymmetric matrix (sometimes referred to as a skew-symmetric matrix) is a square matrix whose transpose equals zero. The matrix (transpose) for every square matrix, A, is provided as AT.

As a result, an antisymmetric matrix or skew-symmetric A can be written as A = -AT. The use of skew-symmetric matrices can be found in a variety of domains, including statistical analysis and machine learning.

Symmetric Matrix

A skew-symmetric and symmetric matrix is both square matrices. The only difference is that a symmetric matrix is equal to its transpose, whereas a skew-symmetric matrix has its transpose equal to its negative. 

It is critical to learn how to find the matrix’s transpose to determine whether it is symmetric. The new matrix is the transpose of the given matrix if we swap the rows and columns of an mxn matrix to get a nxm matrix. 

The number of rows (m) and columns (n) in a matrix can be one of two values:

• The matrices are square if m = n.

• The matrix is rectangular if m ≠ n 

Key pointers:

• A symmetric matrix is defined as a square matrix which is equivalent to its transpose.

• Since only equal matrices have equal dimensions, square matrices are the only ones that are symmetric.

The example of symmetric matrix:

So, A (AT)’s transpose: 

Symmetric Matrix Properties:

• A symmetric matrix is created by adding and subtracting two symmetric matrices.

• The product of A and B is symmetric if A and B are two symmetric matrices that follow the commutative property, i.e. AB =BA.

• If matrix A is symmetric, then matrix An, where n is an integer, is also symmetric.

• A-1 is also symmetric if A is a symmetric matrix.

A matrix that is skew-symmetric

If and only if BT = -B, a square matrix B of size nxn is termed a matrix that is skew-symmetric. A skew-symmetric or antisymmetric matrix is a matrix’s transposed form that is equivalent to the matrix’s negative.

Skew symmetric Matrix Properties:

For being skew-symmetric, the matrix must be square (which is the number of columns and rows must be equivalent) and the matrix is given must match its transpose’s negative. A skew-symmetric matrix has a number of essential characteristics.

• Two symmetric matrices will always provide a skew-symmetric output when added together. With (A+B)T = (A+B), we can obtain the same result as (A+B)T, which is the opposite of what we’d expect.

• The total of all the entries on the major diagonal of a matrix that is skew-symmetric equals zero.

• A matrix that is skew-symmetric A’s real eigenvalue is zero. A matrix’s (skew-symmetric) nonzero eigenvalues are hence unreal.

• When a skew-symmetric matrix is applied to a real or a scalar value, the resulting matrix is also skew-symmetric. The resulting matrix is also skew-symmetric if k is a scalar and B is a skew-symmetric matrix. 

• The matrix I + A, where I is an identity matrix, will be invertible for every real matrix that is skew-symmetric A. 

• Negative semidefinite matrix A, A2 is real skew-symmetric for every real-skew-symmetric semidefinite matrix.

Skew symmetric Matrix Determinant:

A matrix that is skew-symmetric and has an order equivalent to an odd number has a determinant of zero. So, if we come across a skew-symmetric matrix with an odd order, we may simply write 0 as its determinant. The determinant of an odd-ordered matrix that is skew-symmetric is always zero. 

Det (AT) = det (-A) = (-1)n det(A)

Stepwise guide to find a Matrix that is skew-symmetric:

The steps to find a matrix that is skew-symmetric are as follows:

• 1st step: Determine whether the matrix is square, as only square matrices can be deemed skew-symmetric.

• 2nd step: Determine the matrix’s transpose.

• 3rd step: Next, determine the matrix’s negative.

• 4th step: If the matrix’s transpose equals the negative of the provided matrix, the matrix is said to be skew-symmetric.

The eigenvalue of a Matrix that is skew-symmetric:

The eigenvalues of a matrix that is skew-symmetric can only have real or imaginary values, depending on the case. A skew-symmetric matrix’s nonzero eigenvalues are non-real if A has no real eigenvalue.

Note: 

• Any square matrix can be written as the symmetric matrix’s summation and a matrix that is skew-symmetric. 

• When there are real number elements in a matrix (square) called A, the matrix A + AT is symmetric, and the matrix A – AT is a skew-symmetric matrix.

Conclusion

Here’s a rundown of some key topics to remember when studying skew-symmetric matrices.

• Matrixes that have their transposed forms identical to their original forms are known as “skew-symmetric”.

• A symmetric and skew-symmetric matrix can be combined to form any matrix. The square matrix of A, A = A = (1/2)(A + AT) + (1/2)(A – AT). 

• For an odd skew-symmetric matrix, its determinant is zero.