Let $\textbf{A}$ be a symmetric matrix with tthe modified Cholesky factorization $\textbf{A} = \textbf{R}^T\textbf{D}\textbf{R}$ and the classical factorization $\textbf{A} = \textbf{R}_c^T\textbf{R}_c$

Given $\textbf{R}$ and $\textbf{D}$, determine $\textbf{R}_c$



$$ \textbf{D}^{1/2} = \begin{bmatrix} \sqrt{d_1} &&& \\ &\sqrt{d_2}&&\\ &&\ddots&\\ &&&\sqrt{d_n}\ \end{bmatrix} $$

Then multiply each row in the modified Cholesky matrix \textbf{R} by $\sqrt{d_{i,i}}$ to obtain the classical Cholesky matrix

$$ \textbf{R}_c = \textbf{D}^{1/2}\textbf{R} $$

This is confirmed by

\begin{align} \textbf{R}_c^T\textbf{R}_c &= (\textbf{D}^{1/2}\textbf{R})^T(\textbf{D}^{1/2}\textbf{R})\tag{1}\label{eq1} \\ &= \textbf{R}^T\textbf{D}^{1/2}\textbf{D}^{1/2}\textbf{R}\tag{2}\label{eq2} \\&= \textbf{R}^T\textbf{D}\textbf{R}\tag{3}\label{eq3} \end{align}

What is the name of the property of matrix that allow to go from (1) to (2)

  • 1
    $\begingroup$ Distributing a transpose causes order of matrix multiplication to reverse. It's similar to inverse. Also transposing a diagonal matrix does nothing. $\endgroup$ – Mark Dec 27 '18 at 18:43

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