# Diagonalizing matrix with a special conjugate transpose property

Hi all,

I'm looking for the minimum criterion on $A\in M_{3x3}(\mathbb{C})$ (a $3x3$ complex matrix) such that:

1) $A$ is diagonalizable by a matrix $T\in M_{3x3}(\mathbb{C})$

2) $T$ is such that $T^{T}T^{*}=kI$ where $I$ is the 3x3 identity matrix and $k\in\mathbb{C}$

One example of such criterion would be that $Z$ is imaginary with all diagonal entries equal to each other ($x_{ii}=k$) and all off diagonal entries equal to each other ($x_{12}=x_{13}=x_{21}=x_{23}=x_{31}=x_{32}$).

Then, eigenvectors: $\pmatrix{1 & 1 & 1}^T$, $\pmatrix{1 & \alpha^2 & \alpha}^T$, $\pmatrix{1 & \alpha & \alpha^2}^T$ where $\alpha=e^{i2\pi/3}$ with respective eigenvalues: $\lambda_1 = i(a_{ii} + 2x_{ij})$, $\lambda_2 = i(x_{ii} - x_{ij})$, $\lambda_3 = i(x_{ii} - x_{ij})$ can make up a matrix $T$ that satisfies 1 and 2 (with $k$=3)

What is the minimum criterion on $A$ to satisfy these two properties? And how do you approach these kind of problems?

Thanks!

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By $T^*$ do you mean the complex conjugate of $T$? If so, then $T$ is a scalar multiple of a unitary matrix, and the necessary and sufficient condition is that $A$ is normal. – Robert Israel Jun 8 '12 at 21:43
Yes, by $T^*$ I do mean the complex conjugate of $T$. Thanks! – twain Jun 10 '12 at 14:03