Provided that $\mathbf{B}^{\frac{1}{2}}$ exists, how to calculate $\frac{\text{d}\mathbf{x}^{\text{T}}\mathbf{B}^{\frac{1}{2}}\mathbf{x}}{\text{d}\mathbf{B}}$? $\mathbf{x}$ is a column vector and $\mathbf{B}$ is a square matrix.
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The existence of $B^{\frac{1}{2}}$ doesn't imply the existence of $X^{\frac{1}{2}}$ in e neighborhood of $B$, which is required in order to be able to define the derivative. – mathcounterexamples.net Aug 21 '22 at 13:27
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1This related post might help. – greg Aug 21 '22 at 14:01
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@greg Many thanks. – Qiuyun.Zou Aug 21 '22 at 14:12