Hi, Ryota! Thanks for reading it and for the question.

To clarify, we aren’t using PDP^(-1) to compute Z^(T)Z. Rather, we can take our matrix “Z^(T)Z” and decompose it into P, D, and P^(-1). The matrix D is a diagonal matrix that consists of the eigenvalues of Z^(T)Z, the matrix P is the matrix of the eigenvectors of Z^(T)Z, and P^(-1) is the matrix inverse of P. We write that Z^(T)Z = PDP^(-1).

This type of decomposition (or breaking Z^(T)Z into the product of three other matrices) is known as either “the spectral decomposition” or “the eigendecomposition.” You can’t just take any matrix and decompose it this way. However, if you have a matrix that is symmetric and positive semidefinite, then you can use the spectral decomposition!

Because Z^(T)Z will always be symmetric and positive semidefinite, we can always break that matrix out into the product of P, D, and P^(-1).

I hope this helps!