Matrix decomposition

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


 * 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 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) Eigendecomposition of a symmetric matrix: $$A = Q \Lambda Q^T$$
 * 4) 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 $$A^T A$$