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  • Need clarity, kindly explain! If A is Hermitian such that A2 = O, Then

If A is Hermitian such that A2 = O, Then

  • Option 1)

    A=I

  • Option 2)

    A=O

  • Option 3)

    A^3=A

  • Option 4)

    none of these  

 

Answers (1)

best_answer

As we have learned

Herimitian matrices -

A^{\Theta }=A

- wherein

A^{\Theta } is complex conjugate transpose matrix of matrix A

 

  

 

Herimitian matrices -

A^{\Theta }=A

- wherein

A^{\Theta } is complex conjugate transpose matrix of matrix A

 

Let A = [aij]nxn be a Hermitian matrix of order n, so that A^{\theta }=A,

                        i.e. A = \begin{bmatrix} a_1_1 &a_1_2 &.... & a_1_n\\ a_2_1 &a_2_2 &..... &a_2_n \\ .......& ...... &..... &...... \\ a_n_1 &a_n_2 &...... & a_n_n \end{bmatrix} = \begin{bmatrix} \bar{a}_1_1&\bar{a}_2_1 &..... &\bar{a}_n_1 \\ \bar{a}_1_2 &\bar{a}_2_2 & ..... &\bar{a}_n_2 \\ ....... &........ & ........ & ......\\ \bar{a}_1_n& \bar{a}_2_n &..... &\bar{a}_n_n \end{bmatrix}.

                      

                   

Since A2 = O,  each element of AA^{\theta } is zero.

                        \Rightarrow ai1\bar{a}_i_1 + ai2\bar{a}_i_2+ .... + ain\bar{a}_i_n= |ai1|2 + |ai2|2 + .... + |ain|2 = 0

                        \Rightarrow |ai1| = |ai2| = .... = |ain|= 0 \Rightarrow ai1 = ai2= .... = ain= 0.  Hence A = O.

                  

 

 


Option 1)

A=I

Option 2)

A=O

Option 3)

A^3=A

Option 4)

none of these  

Posted by

Aadil

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