- #1

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Find the diagonal form of the Hermitian matrix

[tex]A=\left(

\begin{array}{cc}

2 & 3i\\

-3i & 2

\end{array}

\right)

[/tex]

The spectral theorem could be used with PAP*=D where D is diagonal matrix and P is a unitary matrix.

I put the columns of P as the eigenvectors (with unit length) of A,

[tex]P=\frac{1}{\sqrt{2}}\left(

\begin{array}{cc}

i & -i\\

1 & 1

\end{array}

\right)

[/tex]

I have checked that P is unitary with [tex]P^{-1}=P^{*}[/tex] and the diagonal entries of D should be 5 and -1. But I got

[tex]D=\left(

\begin{array}{cc}

2 & -3\\

-3 & 2

\end{array}

\right)

[/tex]

which clearly isn't correct.

[tex]A=\left(

\begin{array}{cc}

2 & 3i\\

-3i & 2

\end{array}

\right)

[/tex]

The spectral theorem could be used with PAP*=D where D is diagonal matrix and P is a unitary matrix.

I put the columns of P as the eigenvectors (with unit length) of A,

[tex]P=\frac{1}{\sqrt{2}}\left(

\begin{array}{cc}

i & -i\\

1 & 1

\end{array}

\right)

[/tex]

I have checked that P is unitary with [tex]P^{-1}=P^{*}[/tex] and the diagonal entries of D should be 5 and -1. But I got

[tex]D=\left(

\begin{array}{cc}

2 & -3\\

-3 & 2

\end{array}

\right)

[/tex]

which clearly isn't correct.

Last edited: