MATRICES : SOME NEW PROPERTIES. / SB’S THEOREMS (SPECTRUM)

Matrix---not only the arrays of numbers but also has been used as a tool for many calculations in various subjects. Its inverse, eigen values, eigen vectors are of great importance to know its characters. In this paper I have discussed about some new properties of Hermitian, Skew Hermitian matrices, diagonalisation, eigen values, eigen vectors, spectrum, which will open up a new horizon to the students of Mathematics. Also, in this paper I have authored two totally new theorems for students and researchers . SB’s Theorem 1 is on Normality of a block diagonal matrix and SB’s Theorem 2 is on Spectrum of eigen values. These ideas came to me in course of teaching. Hope, these two theorems will be of great help for the students of Physics and Chemistry as well.

A= UDU H , the columns of unitary matrix U are unit eigen vectors and the diagonal elements of diagonal matrix D are the eigen values of A that correspond to those eigen vectors . 6) Among complex matrices ,all unitary , Hermitian and skew Hermitian matrices are normal. However, it is not the case that all normal matrices are either unitary or ( skew)-Hermitian .
Being a student of Mathematics I have tried to share my own thoughts and ideas with other students through this paper . Property 1-If A and B are Hermitian matrices then aA+ bB is also Hermitian for all real scalars a and b .

Proof:
Since A and B are Hermitian matrices we have , Since a ,b are real scalars .

Property 2-A is a Hermitian matrix if and only if iA(or-iA) is skew Hermitian matrix .
Proof: Let A be a Hermitian matrix, then = So , A is Hermitian. Hence proved .
Hence it is proved that B is skew Hermitian matrix .

Property 4-Product of two skew Hermitian matrices A and B is Hermitian if and only if AB=BA Proof:
Here A and b be two skew Hermitian matrices i.e.

Property 5-If A and B are normal matrices with AB=BA then A-B is also normal Proof:
Here we have = , = since A and B are normal .

Property 6-If A be a square matrix of order n then the sum of products of eigen values taken r at a time (r<n) is equal to the [sum of principal minors of A of order r ] .
Proof: Let A be the square matrix [aij] of order n. If 1 , 2 , … … be the eigen values of A, then - The coefficient of − in │ A-λI │ in left hand side is (-1) n [ sum of the principal minors of A of order r ] and the coefficient of − on the right hand side of (ii) is (−1) + [ the sum of the products of eigen values taken r at a time ](r ˂ n) Therefore, the sum of the products of the eigen values taken r at a time = sum of the principal minors of A of order r.

Lemma (1): If A ∈ ℂ × be a block diagonal matrix ,
Then A is diagonalizable if and only if every A ii is diagonalizable for 1 ≤ i ≤ m .

Proof :
The argument is straight forward and is omitted .

Proof:
Let every matrix A ii is normal which automatically implies that each A ii is unitarily diagonalizable. Now , by the lemma 1 , the matrix A is unitarily diagonalizable which in turn implies that the matrix A is normal.