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3 Questions around this concept.
Skew-hermitian matrix
A square matrix is said to be Skew-Hermitian matrix if ∀ i, j,
i .e.
We know that when we take the transpose of a matrix, its diagonal elements remain the same, and while taking conjugate we just change sign from +ve to -ve OR -ve to +ve in imaginary part of all elements, So to satisfy the condition A? = - A, all diagonal element must be purely imaginary. As A? = - A so
Hence all diagonal element should be purely imaginary
E.g
Note:
1. for any square matrix A with elements containing complex numbers, then A-A? is a skew hermitian matrix.
Proof : (A-A?)? = A? - (A?)? = A? - A = -(A-A?), hence skew-hermitian.
2. Every square matrix can be written as the sum of hermitian and skew-hermitian matrix i.e.
If A is a square matrix, then we can write
Properties of hermitian and skew-hermitian matrices
i) If A is a square matrix then AA? and A? A are hermitian matrix.
Proof: for hermitian matrix A? = A, so we check the condition on AA?
(AA?)? = (A?)?A? = AA? hence it is hermitian, and in the same way, A?A will also be hermitian.
ii) If A is hermitian matrix then:
iA is a skew hermitian matrix, where i = √-1
Proof: we need to show (iA)? = -iA
(iA)?= A?i? = A? (-i) = -iA?
Since A is hermitian so A? = A
Hence we have
-iA? = -iA. Proved.
iii) if A is a skew-hermitian matrix, then:
iA is a hermitian matrix, where i = √-1
Proof: we need to show (iA)? = iA
(iA)? = A?i? = A?(-i)
A?(-i) = Ai = iA (since A is skew-hermitian, so A? = -A)
iv) if A and B are hermitian matrices of the same order, then
a. cA and dB are also hermitian matrices of the same order when c and d are scalar real constant.
Since A and B are of the same order, hence they are conformable for addition and by multiplying through a scalar we are just magnifying their values and nothing else, hence they will hold their property of hermitian matrices and cA + dB will be a hermitian matrix.
b. AB is also hermitian if AB = BA
Proof: (AB)? = B?A? = BA = AB (Since A, B are hermitian so A? = A, B? =B)
c. AB + BA will also we hermitian
Proof: from part (b) AB and BA is hermitian and from part (c) AB + BA will also be hermitian.
d. AB - BA will be skew-hermitian
Proof: we need to show (AB-BA)* = -(AB-BA)
(AB-BA)* = (AB)* - (BA)* = B*A* - A*B* = BA - AB = -(AB - BA)
Using A? = A and B? = B, proved.
v) if A and B are skew hermitian matrix then cA +dB will be skew-hermitian
Proof are similar as above, just verify the basic condition, using the given condition of A and B.
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