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Hermitian matrix - Practice Questions & MCQ

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

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Hermitian matrix

Hermitian matrix

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

i .e. \mathrm{A=A^\theta,\;\;[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 and -ve to +ve for imaginary part of all elements, So to satisfy the condition A?  = A diagonal elements must not change, ⇒ all diagonal element must be purely real, 

E.g. 

\\\mathrm{Let,\;\;A=\begin{bmatrix} 3 &3-4i & 5+2i\\ 3+4i& 5 &-2+i \\ 5-2i&-2-i &7 \end{bmatrix}}\\\mathrm{Then,}\\\mathrm{\;\;\;\;\;\;\;\;A'=\begin{bmatrix} 3&3+4i &5-2i \\3-4i &5 &-2-i \\ 5+2i & -2+i &7 \end{bmatrix}}\\\\\\\mathrm{\therefore A^\theta=\overline{(A')}=\begin{bmatrix} 3 &3-4i & 5+2i\\ 3+4i& 5 &-2+i \\ 5-2i&-2-i &7 \end{bmatrix}}\\\\\mathrm{here,\;A\;is\;Hermitian\;matrix\;as\;A=A^\theta}

Note : 

For any square matrix say A,  with complex number entries, 

\\\mathrm{A+A^\theta \;is \;a\;Hermitian\;matrix}\\\mathrm{[\because (A+A^\theta)^\theta=A^\theta+(A^\theta)^\theta=A^\theta+A]}

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Hermitian matrix

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Hermitian matrix Current Topic

Determine The Order Of Matrix
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Matrix Multiplication
Transpose of a Matrix
Symmetric Matrix & Skew Symmetric Matrix
Conjugate of a Matrix
Hermitian matrix
Skew Hermitian Matrix

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