JEE Main Important Formulas 2025 for Physics, Chemistry, Maths

Skew Hermitian Matrix - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

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Skew-hermitian matrix

Skew-hermitian matrix

A square matrix $\mathrm{A=[a_{ij}]_{n\times n}}$ is said to be Skew-Hermitian matrix if $\mathrm{\mathit{a_{ij}=-\overline{a_{ij}}}}$ ∀ i, j,

i .e. $\mathrm{A^\theta=-A,\;\;[where\;A^\theta\;is\;conjugate\;transpose \;of\;matrix \;A]}$

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

$\\\mathrm{a_{ij}=-\overline{a_{ij}} \;\; \forall \;i,j} \\\mathrm{if \; we \; put \; i=j, \; we\; have} \\\mathrm{a_{ii}= -\overline{a_{ii}} \Rightarrow a_{ii}+\overline{a_{ii}}=0} \\\mathrm{\Rightarrow a_{ii}=0}$

Hence all diagonal element should be purely imaginary

E.g

$\\\mathrm{Let,\;\;A=\begin{bmatrix} -3i &-3-4i & -5+2i\\ 3-4i& 5i &i \\ 5+2i&i &0 \end{bmatrix}}\\\mathrm{Then,}\\\mathrm{\;\;\;\;\;\;\;\;A'=\begin{bmatrix} -3i&3-4i &5+2i \\-3-4i &5i &i \\ -5+2i & i &0 \end{bmatrix}}\\\\\\\mathrm{\therefore A^\theta=\overline{(A')}=\begin{bmatrix} 3i&3+4i &5-2i \\-3+4i &-5i &-i \\ -5-2i & -i &0 \end{bmatrix}}\\\\\\\mathrm{\;\;\;\;\;\;\;\;=-\begin{bmatrix} -3i &-3-4i & -5+2i\\ 3-4i& 5i &i \\ 5+2i&i &0 \end{bmatrix}=-A}\\\\\mathrm{here,\;A\;is\;Skew-Hermitian\;matrix\;as\;A^\theta=-A}$

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 $\\\mathrm{A = \frac{1}{2}(A + A^{\theta})+\frac{1}{2}(A-A^{\theta})}$

Properties of hermitian and skew-hermitian matrices

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.