# Compare classical and modified algorithm of Cholesky

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 $$\textbf{R}_c$$

$$\textbf{Proof}$$:

Let

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

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)

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