Special Matrices

 Symmetric and Skew Matrices

A square matrix A = [a_ij] is said to be symmetric when a_ij = a_ij for all i and j. If a_ij = -a_ij for all i and j and all the leading diagonal elements are zero, then the matrix is called a skew symmetric matrix.

For example:

is a symmetric matrix and is a skew-symmetric matrix.

Hermitian and Skew - Hermitian Matrices

A square matrix A = [a_ij] is said to be Hermitian matrix if a_ij = _ij, i, j i.e. A^θ = -A.

For example:

        are Hermitian matrices.

Note:   
 * If A is a Hermitian matrix then a_ii =_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 = [a_ij] is said to be a skew-Hermitian if a_ij =
-_ij, i, j i.e. A^θ = -A.

For example:

        are skew-Hermitian matrices. 

Note:
 * If A is a skew-Hermitian matrix then a_ii =_ij => a_ii + _ii = 0, i.e. a_ii 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.

 Unitary Matrix

A square matrix is said to be unitary if A = I. Since || = |A| and |A|=||, we have |||A| = 1.

Thus the determinant of a unitary matrix is of unit modulus. For a matrix to be unitary it must be non-singular.

Hence, A = I => A = I.

Orthogonal Matrix

Any square matrix A of order n is said to be orthogonal if AA' = A'A = I_n.

Idempotent Matrix

A square matrix A is called idempotent provided it satisfies the relation A² = A.

For example:

        The matrix A = is idempotent as

        A² = A.A = = A.

Involuntary Matrix

A matrix such that A² = I is called involuntary matrix.

Nilpotent Matrix

A square matrix A is called a nilpotent matrix if there exists a positive integer m such that A^m = O. If m is the least positive integer such that  A^m = O, then m is called the index of the nilpotent matrix A

Illustration:

Suppose a, b, c are real numbers such that abc = 1. If A = is such that A'A = I, then find the value of a³ + b³ + c³.

Solution:

          Note A' = A.

        Thus, I = A'A = AA = A²

        |A²| = |A²| = |I| = 1

        |A| = + 1. But |A| = a³ + b³ + c³ - 3abc.

        Thus, a³ + b³ + c³ - 3abc = + 1

        => a³ + b³ + c³ = 4, 2.

 Illustration:

If ω ≠ 1 is a cube root of unity, then show that

A =is singular matrix.

Solution:

       

Hence A is singular matrix.

Illustration:

Show that the matrix = is nilpotent matrix of index 3.

 Solution:

               

          => A³ = 0 i.e. A^k = 0. Here k = 3.

        Hence A is nilpotent matrix of index 3.

Adjoint of a Square Matrix

Let A = [a_ij] be a square matrix of order n and let C_ij be cofactor of a_ij in A. Then the transpose of the matrix of cofactors of elements of A is called the adjoint of A and is denoted by adj A.

where C_ij denotes the cofactor of a_ij in A.

For example:  A = , C₁₁ = s, C₁₂ = -r, C₂₁ = -q, C₂₂ = p

                     => adj A = .

Theorem:  Let A be a square matrix of order n. Then A(adj A) = |A| I_n = (adj A)A.

Proof:      Let A = [a_ij], and let C_ij be cofactor of a_ij in A. Then

               (adj A)_ij = C_ij 1 < i, j , n.

                 Now, (A(adj A))_ij = ∑ⁿ_r=1(A)_ir  (adj A)_rj = ∑ⁿ_r=1 a_ij C_jr 

                 => A (adj A) = = |A| I_n.

              Similarly ((adj A)A)_ij= ∑ⁿ_r=1(adj A)_ir (A)_rj= ∑ⁿ_r=1C_ri a_rj=

               Hence, A(adj A) = |A| I_n = (adj A)A.

 Note : The adjoint of a square matrix of order 2 can be easily obtained by interchanging the diagonal elements and changing the signs of off-diagonal (left hand side lower corner to right hand side upper corner) elements.

Inverse of a Matrix

A non-singular square matrix of order n is invertible if there exists a square matrix B of the same order such that AB = I_n = BA.

In such a case, we say that the inverse of A is B and we write, A^-1 = B.

The inverse of A is given by A^-1 = 1/|A|. adj A.

Properties of Inverse of a Matrix

(i) Every invertible matrix possesses a unique inverse.

(ii) (Reversal law) If A and B are invertible matrices of the same order, then AB is invertible and (AB)^-1 = B^-1 A^-1.

       In general,if A,B,C,...are invertible matrices then (ABC....)^-1 =..... C^-1 B^-1 A^-1.

(iii) If A is an invertible square matrix, then A^T is also invertible and (A^T)^-1 = (A^-1)^T.

(iv) If A is a non-singular square matrix of order n, then |adj A| = |A|^n-1.

(v) If A and B are non-singular square matrices of the same order, then adj (AB) = (adj B) (adj A).

(vi) If A is an invertible square matrix, then adj(AT) = (adj A)T.

(vii) If A is a non-singular square matrix, then adj(adjA) = |A|^n-1A.

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