Symmetric Lower Upper Triangular Factorizations

Why

What of lower upper triangular factorizations when the system is symmetric?

Definition

A symmetric lower upper triangular factorization of $A$ is a pair of matrices $(L \in \R ^{m \times m}, U \in \R ^{m \times m})$ where $L$ is unit lower triangular, $U$ is upper triangular and $A = LU$.

\[ A = LL^\top . \]

Other terminology includes lower upper triangular factorization, LU decomposition, LU factorization. Define $R = L^\top $, then

\[ A = R^\top R. \]

Let $A \in \R ^{n \times n}$. Then the ordinary row reduction of $A$ is a matrix $U$ which is upper triangular.be symmetric. A lower upper triangular decomposition $A$ is a pair of matrices $(L, \transpose{L})$ where $L \in \R ^{n \times n}$ is lower triangular, has nonnegative real diagonal entries, and satisfies

\[ A = LL^\top . \]

Other terminology includes lower upper triangular factorization, LU decomposition, LU factorization. Define $R = \transpose{L}$, then

\[ A = \transpose{R}R. \]

Basic properties

Let $A \in \R ^{m \times m}$ be positive definite. Then there exists unique lower triangular matrix $L \in \R ^{n \times n}$ so that

\[ A = L\transpose{L}. \]

So, in the case that $A$ is positive definite, a lower upper triangular decomposition exists and is unique. Therefore we refer to it as the upper lower triangular decomposition of $A$. It is also known (universally) as the Cholesky decomposition or Cholesky factorization of $A$.

If $A$ is positive semidefinite, there exists a permutation matrix $P$ for which there is a unique $L$ so that

\[ P^\top AP = LL^\top . \]

This second proposition says that there is a unique Cholesky decomposition for a particular permutation (or pivoting) of $A$.

Unitriangular form

A lower diagonal upper decomposition (or lower diagonal upper factorization) of a matrix $A$ a sequence $(L, D, \transpose{L})$ where $L \in \R ^{n \times n}$ is unit lower triangular, $D \in \R ^{n \times n}$ is diagonal with real nonnegative entries and

\[ A = LDL^\top . \]

Other terminology includes LDL decomposition, LDL factorization, LDU factorization, LDU decomposition.

If $(L \in \R ^{n \times n}, D \in \R ^{n \times n}, \transpose{L})$ is a LDU decomposition of $A \in \R ^{n \times n}$, then $(L\Msqrt{D}A = LD\transpose{L}$ then $(\tilde{L}\Msqrt{D}, \Msqrt{D}\transpose{L})$ is a LU decomposition. Conversely, if $(B, \transpose{B})$ is a LU decomposition and $S$ is the diagonal matrix satisfying $S_{ii} = B_{ii}$ for $i = 1, \dots , n$, then $(B\inv{S}, S^2, \inv{S}\transpose{B})$ is a LDU decomposition of $A$.