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I need a help please when I have a matrix with complex eigenvalues for example $$A=\begin{pmatrix}0& 1& 0\\0 &0 &1\\-24 &-29 &-18\end{pmatrix}$$ with its eigenvalues of: $-16.3$, $-0.844+0.871j$, and $-0.844-0.871j$

Matrix $A$ can be diagonalized to the classical known form of $$A_\text{diag}=\begin{pmatrix}-0.844+0.871j& 0& 0\\0 &-0.844-0.871j& 0\\0 &0 &-16.3\end{pmatrix}$$ with a vandermone transformation matrix

The Question is I need a transformation matrix to transform matrix $A$ to the form of matrix $A_d$ (with no complex elements) $$A_d=\begin{pmatrix}-0.844 &0.871& 0\\-0.871 &-0.844& 0\\0 &0 &-16.3\end{pmatrix}$$ and this form called a quasi-diagonal matrix

Thank you

lioness99a
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Ashraf Abdulhamed
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1 Answers1

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$$ \frac{1}{2} \; \left( \begin{array}{rr} -i & 1 \\ 1& -i \end{array} \right) \left( \begin{array}{rr} i &1 \\ 1& i \end{array} \right) = \left( \begin{array}{rr} 1&0 \\ 0&1 \end{array} \right) $$

$$ \frac{1}{2} \; \left( \begin{array}{rr} -i & 1 \\ 1& -i \end{array} \right) \left( \begin{array}{cc} a + b i&0 \\ 0&a - bi \end{array} \right) \left( \begin{array}{rr} i &1 \\ 1& i \end{array} \right) = \left( \begin{array}{rr} a&b \\ -b&a \end{array} \right) $$

Will Jagy
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