Article Type: Research Article

 

Article Citation: Surajit Bhattachaaryya. (2021). MATRICES : SOME NEW PROPERTIES. /  SB’S THEOREMS (SPECTRUM). International Journal of Research -GRANTHAALAYAH, 9(2), 181-186. https://doi.org/10.29121/granthaalayah.v9.i2.2021.3408

 

Received Date: 01 February 2021

 

Accepted Date: 28 February 2021

 

Keywords:

Hermitian

Skew Hermitian

Normal

Diagonalizable Matrix

Eigen Values

Block Diagonal Matrix

Spectrum
ABSTRACT

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.

1.     INTRODUCTION

 

Recall

1)     A complex square matrix A will be Hermitian if it’s (rs)th element and (sr)th element be complex conjugate of one another i.e. A = [ Ā ] ^T =A ^H .

2)     The entries on the main diagonal of any Hermitian matrix are real .

3)     A complex square matrix A is Unitary if  AA^H = I .

4)     A complex square matrix A is normal iff  A^HA = AA^H.

5)     A matrix  A ∈   is normal  if and only if it is unitarily diagonalizable  i.e.

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 ,          

                    

                                                                                                                                                

 

Now,          

 

i.e           

     

 Since  a  ,b are real scalars .

     

 i.e .

 

It proves that   

 

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   

 

Which  proves  that  iA  is  skew  Hermitian .

Conversely,             let           ,

                        or ,     

                        Or ,     A^H = iA

                        or ,         A^H = A . 

                    

So , A is Hermitian.

                                             Hence proved .

 

Property 3- The inverse of an invertible   skew  Hermitian matrix is  skew Hermitian.

 Proof :  Let A be an invertible skew Hermitian matrix i.e. A^H = - A and if B be the inverse of A then A.B = I (Identity matrix) i.e . B = A^—1 . Now B ^H =( A^—1 ) ^H = ( A ^H )^—1

 

             =( ̶  A )^—1 =  ̶  A^—1 =  ̶  B .

 

 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.  

Now,          

                    

Thus ,

                                    Hence  proved.

 

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 .

 

 

 

Now,             
                                    =
                                    = 
                                    = 

 

Note:          Fuglede’s Theorem :  If bounded operator T and S on a Hillbert space commute  and S is normal  then T andcommute .

 

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  be the eigen values of A,                      

then –

 

 ………………(ii)

 

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  [ 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.

Hence proved .

 

      Lemma (1): If A ∈  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 .

 

SB’ s Theorem ( 1 ) :     Statement :

Let    A ∈    be a block diagonal matrix

 

 

 

 

 

 

 

 

 

 

Then A is normal if and only  if every matrix A_ii  is normalfor  1≤ i ≤ m .

 

 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.

 

The proof of the converse implication is very simple and so left for the reader.  (proved)

 

 Lemma (2):  Let A be a Hermitian matrix and  λ₁  ≥  λ₂  ≥ ------≥ λ_n  be its eigen values with the orthonormal  eigen vectors u₁ ,u₂ ,------- u_n  respectively . Let us define a subspace

 

 

Proof :       Let  x be a unit vector in S , then    for p ≤ i ≤ q .Since  ∥x∥₂ =1, we have ∣v_p∣ ²  + ------+ ∣v_q∣ ² = 1 . This allows us to write -----------------

x^HAx =  x^H(v_p Au_p + -------------- + v_q Au_q)

          = x^H(v_p λ_p u_p + ------------ + v_q λ_q u_q)

         =  x^H( ∣v_p∣ ² λ_p + ---------- + ∣v_p∣ ² λ_q).

    Since    ∣v_p∣ ²  + ------+ ∣v_q∣ ² = 1 , the desired inequality  follows .

 

SB’s  Theorem ( 2 ) (spectrum) :

 

Let   A ,B ∈   be two Hermitian matrices .  Define , E =A+B .  Suppose that the eigen values of A ,  B , E are

 ------- ≥  and  respectively .

 

Then we have        

 

Proof :       Since A and B are two Hermitian  matrices , E is also Hermitian. So all matrices involved have real eigen values .

 

By Courant –Fisher Theorem--------------

 

 

 

 

 

Let us take unitary matrix U such that  U^HAU = diag ( α₁ ,α₂ ,-------------- α_n ).

Let  w_i = Ue_i ,for 1 ≤ i ≤ k-1 . Then we have ------------

 

    w_i^Hx  =  e_i^HU^Hx  =  0    for 1 ≤ i ≤ k-1 .

 

Let us define  Y = U^Hx  and since U is an unitary matrix ,∥ Y∥ ₂ = 1  and  ∥ x∥ ₂ = 1  ,

We have    e_i^HY = y_i = 0   for  1 ≤ i ≤ k .

Therefore  . This in turn implies that

 

x^HAx  = Y^HU^HAUY =   .

 

Following the same process we can show that x^HBx ≤

So , from (iii)  we have   +  -----------------------(iv)

               

By the lemma (2) and from inequality –(iv)  we can conclude the desired inequality

            

 Ɛ_n ≤ + α_i ≤ Ɛ₁  .       

                                       Hence  proved  .

 

SOURCES OF FUNDING

 

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

 

CONFLICT OF INTEREST

 

The author have declared that no competing interests exist.

 

ACKNOWLEDGMENT

 

None.

 

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