If A is a square matrix then A-AT is a skew hermitian matrix-

Lets take general element from matrix A and verify if A A^T is a skew symmetric matrix. Assume aij = p + iq aji = r + is The expression A A^T will translate ...

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Standard XIII

Mathematics

Skew Hermitian Matrix

If A is a squ...

Question

If A is a square matrix then $A -^{T}$ is a skew hermitian matrix.

True

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False

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Solution

The correct option is A
True

Lets take general element from matrix A and verify if $A -^{T}$ is a skew symmetric matrix.

Assume

aij = p + iq

aji = r + is

The expression $A -^{T}$ will translate to aij $= - ¯ ¯¯¯¯¯ ¯ a j i$ at the elementary level in the resultant matrix.

If mij is the ijth element in resultant matrix

mij = $a i j - ¯ ¯¯¯¯¯ ¯ a i j = (p + i q) - (r - i s)$

=(p – r)+ i (q + s) ….(1)

mji = $a i j - ¯ ¯¯¯¯¯ ¯ a i j = (r + i s) - (p - i q)$

= r – p + I (q+s) ….(2)

From (1) & (2) we can see that

$m i j = - ¯ ¯¯¯¯¯¯¯ ¯ m j i$

Thisimplies that the resultant matrix or $A -^{T}$ is skew hermitian.

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In a square matrix of order 4, the element $a_{33}$ = 33. This square matrix cannot be skew hermitian.

Q. A square matrix A such that

A^{θ} = A

is called Hermitian matrix i.e.

a_{i j} =_{j i}

for all values of i and j and a square matrix A such that

A^{θ} = - A

is called skew –Hermitian matrix i.e.

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\to

{1,-1}, M is set of all hermitian or Skew–hermitian matrices, be a function defined as

f (A) = {\begin{matrix} 1 A i s h e r m i t i a n - 1 A i s s k e w h e r m i t i a n \end{matrix}

f (A - A^{θ}) =

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If A is a square matrix then A−¯¯¯¯¯¯¯AT is a skew hermitian matrix.

If A is a square matrix then $A -^{T}$ is a skew hermitian matrix.