Making a matrix Toeplitz

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If $A$ is a Hermitian square matrix, then can we say the following is a Toeplitz matrix?

$$\mathrm{diag}(A)^{-1/2} A\, \mathrm{diag}(A)^{-1/2}$$

Or, in other words, what condition $A$ has to satisfy so the resulting would be Toeplitz?