Question 21046: Matrices with the property A*A=AA* are said to be normal.
1. Verify that symmetric matrices and hermitian matrices are normal.
2. Give and example of a 2x2 matrix which is not symmetric nor hermitian but normal
3. Prove that if A is normal, then R(A) _|_ N(A).
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! Matrices with the property A*A=AA* are said to be normal.
1. Verify that symmetric matrices and hermitian matrices are normal.
2. Give and example of a 2x2 matrix which is not symmetric nor hermitian but normal
3. Prove that if A is normal, then R(A) _|_ N(A).
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I WOULD LIKE TO SUGGEST A FEW THINGS FIRST..............
1.PLEASE TYPE YOUR PROBLEM CORRECTLY.IF THERE IS PROBLEM IN TYPING ,COPY IT IN HAND WRITING/SCAN IT OR DESCRIBE FULLY IN WRITING.WE ARE NOT TAKING UP MANY PROBLEMS BECAUSE THE PROBLEMS AS TYPED ARE NOT COHERENT..FOR EXMPLE HERE A*A=AA* DOES NOT MEAN ANY THING.IT IS NOT CORRET DEFIITION AT ALL...WE HAVE TO INFER FROM YOUR DESCRIPTION THAT THEY ARE NORMAL.
2.IT WOULD HELP IN OUR EXPLANATION IF WE KNOW YOUR BASIC STANDARD AND PRESENT COURSE OF STUDY..SOLUTION ALSO DIFFERS .FOR EXAMPLE IN THIS QUESTION YOUWANTED EXAMPLES OF 2X2 MATRICES AND NOT GENERAL ANY ORDER MATRICES INDICATING ELEMENTARY APPROACH .WE CAN GIVE EXPLANATION AND ANSWER BASED ON THAT.
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OK...NOW TO THE PROBLEM ..FIRSTLY NORMAL MATRICES..DEFINITION..ONE MATRIX A IS CALLED NORMAL IF A AND THE TRANSPOSE OF ITS COMPLEX CONJUGATE COMMUTE WITH EACH OTHER..OR EXPRESSED IN SYMBOLS
A*((A")')=((A")')*A..WHERE
A=GIVEN SQUARE COMPLEX MATRIX.....IMPORTANT A SHOULD BE SQUARE MATRIX.
A"=CONJUGATE OF A..
A' =TRANSPOSE OF A
* DENOTES MULTIPLICATION
MATRIX MLTIPLICTION
IS IN GENERAL NOT COMMUTATIVE EXCEPT IN ONLY FEW SPECIAL CASES LIKE A AND ITS INVERSE ETC...
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1.LET US TAKE A SYMMETRIC MATRIX..A....OF 2ND.ORDER..(HERE ONY I WANT TO KNOW WHETHER YOU WANT PROOF FOR A GENERAL ORDER OR ONLY SECOND ORDER
x1+iy1........x2+iy2
x2+iy2........x1+iy1
is matrix A
its conjugate A" is given by
x1-iy1........x2-iy2
x2-iy2........x1-iy1
its transpose (A")' is same as A" above since A" is symmetric
so we have to prove that
A*A" = A"*A which you can eaily verify by multiplication...general proof for any order can also be built up on the same basis...try and come back if you dont get it.....
also if you are familiar with symbolic notation of a matrix elements by Aij...then this can be easily proved in those general terms for any order matrix...try it if you are conversant with that notation.
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now on to a hermitian matrix...let A be a hermitin matrix given by
As you know A is square matrix and is called hermitian if (A")'=A
IMPORTANT POINT HERE IS THAT IF A IS HERMITIAN THEN IN THE STANDARD NOTATION
Aij=(Aji)" FOR ALL VALUES OF i AND j...AND HENCE DIAGONAL ELEMENTS WILL ALL BE REAL SINCE Aii=(Aii)"
so we have to prove that
A*((A")')=((A")')*A..but by definition of hermitian matrix as given above,we have the
l.h.s=A*A and r.h.s = A*A..so a hermitian matrix is normal
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you can note this theorem on normal matrices...
A SQUARE MATRIX IS NORMAL IF AND ONLY IF , IT CAN BE EXPRESSED AS H+iK WHERE H AND K ARE 2 HERMITIAN MATRICES WHICH COMMUTE WITH EACH OTHER...
TAKING H AS
1.....i
-i....1
AND K AS
1......-i
i......1
WHICH ARE HERMITIAN AND COMMUTATIVE..SINCE HK=KH=
0....0
0....0
WE GET A 2X2 NORMAL MATRIX AS H+iK=
1+i.......1+i
-1-i......1+i
WHICH IS NEITHER SYMMETRIC NOR HERMITIAN
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NOW THE HIRD QUESTION AGAIN TYPING IS NOT CLEAR PLEASE EXPLAIN WHAT IS
R(A) _|_ N(A).....WHAT IS THAT SYMBOL IN THE MIDDLE ,WHAT IS R(A) AND WHAT IS
N(A) SO THAT WE CAN GIVE THE SOLUTION
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