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I am very confused by a line in the textbook "Gaussian Processes for Machine Learning" by Rasmussen and Williams (available here). On page 48, they write the following:

$|B| = |K| \cdot |K^{-1} + W| = |I_n + W^{\frac{1}{2}}KW^{\frac{1}{2}}|$

I don't understand how the second equality follows. I understand that the following works:

$|K| \cdot |K^{-1} + W| = |I_{n} + KW|$

due to the properties of determinants, but I just can't work out how $KW = W^{\frac{1}{2}}KW^{\frac{1}{2}}$.

More information that may be important:

  • $W$ is a diagonal matrix whose entries are all positive
  • $K$ is a symmetric positive definite matrix.

Thanks for your help!

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The diagonal entries of $W$ are positive, $W$ has a square root that is invertible.

$$|I+KW|=|W^{-\frac12}+KW^\frac12||W^\frac12|=|W^\frac12||W^{-\frac12}+KW^\frac12|=|I+W^\frac12KW^\frac12|$$

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