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 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. INTRODUCTIONRecall 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 AAH = I . 4) A complex square matrix A is normal iff AHA = AAH. 5) A matrix A ∈ is normal if and only if it is unitarily diagonalizable i.e. A= UDUH, 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 , AH = 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. AH = -
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 Aii 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 Aii is normalfor 1≤ i ≤ m . Proof: Let every matrix Aii is normal which automatically implies that each Aii 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 ≥
------≥ λn be its eigen values with the orthonormal eigen vectors u1 ,u2
,------- un 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 ∣vp∣ 2 + ------+ ∣vq∣
2 = 1 . This allows us to write ----------------- xHAx = xH(vp Aup
+ -------------- + vq Auq) = xH(vp λp
up + ------------ + vq
λq uq) =
xH( ∣vp∣
2 λp + ---------- +
∣vp∣ 2 λq). Since ∣vp∣
2 + ------+ ∣vq∣ 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 UHAU = diag ( α1 ,α2
,-------------- αn ). Let
wi = Uei
,for 1 ≤ i ≤ k-1 . Then we have ------------ wiHx
= eiHUHx =
0 for 1 ≤ i ≤ k-1 . Let us define Y = UHx and since U is an
unitary matrix ,∥ Y∥
2 = 1
and ∥
x∥ 2 = 1 , We have
eiHY = yi = 0
for 1 ≤ i ≤ k . Therefore
.
This in turn implies that xHAx = YHUHAUY = . Following
the same process we can show that xHBx
≤ So , from (iii) we have
+ -----------------------(iv) By the
lemma (2) and from inequality –(iv) we
can conclude the desired inequality Ɛn ≤ + αi ≤ Ɛ1 .
Hence proved . SOURCES OF FUNDINGThis research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors. CONFLICT OF INTERESTThe author have declared that no competing interests exist. ACKNOWLEDGMENTNone. REFERENCES [1] Das.A. N. Advanced Higher Algebra .3Edn
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