Question

If A is hermitian then iA is skew hermitian

Answer 1

$\large\underline{\sf{Solution-}}$

Given that

$\rm :\longmapsto\:A \: is \: hermitian \: matrix.$

We know,

A square matrix A is said to be Hermitian iff

$\rm :\longmapsto\:\boxed{ \tt{ \: {A}^{\theta } = A \: }}$

and

A square matrix A is said to be skew - hermitian iff

$\rm :\longmapsto\:\boxed{ \tt{ \: {A}^{\theta } \: = \: - \: A \: }}$

It is given that

$\rm :\longmapsto\:A \: is \: hermitian \: matrix.$

$\rm :\longmapsto\:\boxed{ \tt{ \: {A}^{\theta } = A \: }}$

Now, Consider

$\rm :\longmapsto\: {(iA)}^{\theta }$

We know,

$\boxed{ \tt{ \: \overline{i} = - \: i \: }}$

So, using this, we get

$\rm \: = \: - \: i \: {A}^{\theta }$

$\rm \: = \: - \: i \: A$

$\rm \implies\: {(iA)}^{\theta } \: = \: - \: i \: A$

$\bf\implies \:iA \: is \: skew - hermitian$

Hence, Proved

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Additional Information :-

1. If A is hermitian, then main diagonal entries are purely real

2. If A is skew - hermitian matrix, then main diagonal entries are purely imaginary.

Answer 2

Answer:

$\large\underline{\sf{Solution-}}$

Given that

$\rm :\longmapsto\:A \: is \: hermitian \: matrix.$

We know,

A square matrix A is said to be Hermitian iff

$\rm :\longmapsto\:\boxed{ \tt{ \: {A}^{\theta } = A \: }}$

and

A square matrix A is said to be skew - hermitian iff

$\rm :\longmapsto\:\boxed{ \tt{ \: {A}^{\theta } \: = \: - \: A \: }}$

It is given that

$\rm :\longmapsto\:A \: is \: hermitian \: matrix.$

$\rm :\longmapsto\:\boxed{ \tt{ \: {A}^{\theta } = A \: }}$

Now, Consider

$\rm :\longmapsto\: {(iA)}^{\theta }$

We know,

$\boxed{ \tt{ \: \overline{i} = - \: i \: }}$

So, using this, we get

$\rm \: = \: - \: i \: {A}^{\theta }$

$\rm \: = \: - \: i \: A$

$\rm \implies\: {(iA)}^{\theta } \: = \: - \: i \: A$

$\bf\implies \:iA \: is \: skew - hermitian$

Hence, Proved

▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬

Additional Information :-

1. If A is hermitian, then main diagonal entries are purely real

2. If A is skew - hermitian matrix, then main diagonal entries are purely imaginary.