Matrix Factorization

A matrix factorization (or matrix decomposition) is the right-hand-side product in

for “input” matrix . The number of factor matrices depends on the situation. Most often, or .

Note that the process of producing a factorization/decomposition is also called “factorization” or “decomposition”.

Examples

Some common factorizations and related devices are:

• LU-decomposition: , where is lower triangular, and is upper triangular
• QR-decomposition: , where is orthogonal, and is right triangular.
• Singular value decomposition (SVD): , where and are orthogonal, and is a partially diagonal matrix.
• The Cholesky Decomposition.
• For a positive definite matrix, we can decompose it into its square root squared.
• Polar decomposition
• Jordan canonical form
• Iwasawa decomposition

See the entries for these and other matrix factorizations for details on the contents of the factor matrices, where to apply them, and how to best calculate them.

Simultaneous matrix factorization

• commuting matrices are simultanenously triangularizable
• commuting normal matrices are simultanenously diagonalizable