Matrix decomposition

There are a lot of interesting matrix decompositions. Below are listed some of them.

Exact decompositions

 * 1) QR Factorization: When we take a full-rank matrix $$A$$ and compute an orthogonal basis ($$Q$$) and an upper triangular matrix ($$R$$) that gives the coefficients of the linear combination of the columns of $$Q$$ that recover $$A$$.
 * 2) Eigendecomposition of a square and diagonalizable matrix, $$A = T \Lambda T^{-1}$$ where $$\Lambda$$ is a diagonal matrix that contains eigenvalues and $$T$$ is a matrix that contains eigenvectors.
 * 3) Triangularization of a square matrix is finding an orthogonal matrix $$Q$$ such that for any matrix $$A$$ the matrix $$T=Q'AQ$$ is upper triangular with eigenvalues of $$A$$ on its diagonal. (see also Schur's triangularization theorem)
 * 4) Jordan Canonical Form of any square matrix $$A = T J T^{-1}$$, which is a generalization of diagonalization where $$J$$ is a bidiagonal matrix.
 * 5) Eigendecomposition of a symmetric matrix: $$A = Q \Lambda Q^T$$
 * 6) Singular Value Decomposition of any matrix $$A$$ into $$A = U \Sigma V^T$$, where $$\Sigma$$ is the diagonal matrix that contains the square-roots of the eigenvalues of $$A A^T$$, the columns of $$U$$ are eigenvectors of $$AA^T$$ and columns of $$V$$ are eigenvectors $$A^T A$$
 * 7) Full-rank factorization of any $$m\times n$$ matrix $$A$$ as $$A=BC$$ into two matrices $$B, C$$ such that $$B \in \mathbb{R}^{m\times r}$$ and $$B \in \mathbb{C}^{r\times n}$$, where $$r=\text{rank}(A)$$.

The following are typical decompositions that are used for (computationally) simplifying the solving of $$Ax=b$$
 * 1) LU decomposition for any nonsingular matrix
 * 2) Cholesky factorization for any symmetric positive definite matrix
 * 3) LDL factorization for any symmetric nonsingular matrix

Approximate decompositions

 * 1) Using SVD one can approximate any matrix $$A$$ using $$p$$ singular values as: $$A \approx U_1 \Sigma_1 V^T_1$$.