Factoring a matrixAn $m$ by $n$ matrix $A$ defines a linear transformation from $R^m$ to $R^n$ using right multiplication $A\ x=y$. There are many factorizations of $A$ that have been developed over the years. The PLU factorization arose from Gaussian elimination as an efficient way to solve systems of linear equations with the same matrix $A$ of coefficients. The Cholesky factorization was designed to efficiently solve systems of linear equations with a symmetric matrix $A$ of coefficients. The QR factorization writes $A = QR$, where $Q$ is an orthonormal matrix and $R$ is an upper triangular matrix. The SVD (singular value decomposition) of $A$ is $A=V S Uh^T$ where $V$ and $Uh$ are orthonormal matrices and $S$ is a diagonal matrix with nonincreasing diagonal entries. It can be used to find the 'best' solution of an underdetermined system of equations or an overdetermined system of equations. The following interacts allow visual investigation some of the various standard factorizations of a matrix. Press this button first to start or restart everything. 

The Singular Value Factorization for 2 by 2 matrices. 
Singular Value factorization for 3 by 3 matrix. 
PLU factorization for 3 by 3 matrix. 
QR factorization for 3 by 3 matrix. 