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Is there any unitary transformation that has the effect of only multiplying the diagonal with some values.

For example if I start with the matrix

$$A=\left(\matrix{1&a&b&c\\d&1&e&f\\g&h&1&i\\j&k&l&1}\right)$$

Is there some unitary matrix which will transform it to

$$A=\left(\matrix{\alpha&a&b&c\\d&\beta&e&f\\g&h&\gamma&i\\j&k&l&\delta}\right)$$

where we have some target $\{\alpha,\beta,\gamma,\delta\}$.

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  • $\begingroup$ How are you going to apply the unitary transformation to $A$? Are you regarding the matrix as a linear transformation and doing $A U$, or are you regarding it as an inner product and doing $U^* A U$? $\endgroup$ – Daniel McLaury Feb 22 at 22:41
  • $\begingroup$ @DanielMcLaury I intended it to be in the form $U^*AU$ $\endgroup$ – James Feb 22 at 22:41
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Suppose $A = I$, the identity matrix. Then your condition says that $U^* U$ is equal to some non-identity matrix, contradicting the definition of a unitary matrix.

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