An $m$ by $n$ matrix of real numbers $A$ 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 non-increasing diagonal entries. It can be used to find the 'best' solution of an under-determined system of equations or an over-determined 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.