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>> Special Matrices Part-1
Symmetric and Skew Matrices
A square matrix A = [aij] is said to be symmetric when aij = aij for all i and j. If aij = -aij for all i and j and all the leading diagonal elements are zero, then the matrix is called a skew symmetric matrix.
is a symmetric matrix and is a skew-symmetric matrix.
Hermitian and Skew - Hermitian Matrices
A square matrix A = [aij] is said to be Hermitian matrix if aij = ij, i, j i.e. Aθ = -A.
are Hermitian matrices.
* If A is a Hermitian matrix then aii =ij is a real i. Thus every diagonal element of a Hermitian Matrix must be real.
* A Hermitian matrix over the set of real numbers is actually a real symmetric matrix.
And a square matrix, A = [aij] is said to be a skew-Hermitian if aij =
-ij, i, j i.e. Aθ = -A.
are skew-Hermitian matrices.
* If A is a skew-Hermitian matrix then aii =ij => aii + ii = 0, i.e. aii must be purely imaginary or zero.
* A skew-Hermitian Matrix over the set of real numbers is actually a real skew- symmetric matrix.
Singular and Non-singular Matrices
Any square matrix A is said to be non-singular if |A| ≠ 0, and a square matrix A is said to be singular if |A| = 0. Here |A| (or det(A) or simply det A) means corresponding determinants of square matrix A e.g. if
10 - 12 = -2 => A is a non-singular matrix.