Singular value decomposition

Unitary or orthogonal factors and a rectangular diagonal factor describe any finite matrix.

A=U\Sigma V^*

Different ways to understand it

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Pseudoinverse
The Moore–Penrose inverse generalizes matrix inversion to least-squares and rank-deficient problems.
Singular values
Singular values are nonnegative square roots of eigenvalues of the conjugate-transpose product.
Basis and dimension
A basis is a linearly independent spanning set; every vector has unique coordinates in that basis.
Condition numbers
A matrix condition number describes sensitivity of linear-system solutions to perturbations.
Determinants
The determinant of a square matrix tracks signed volume scaling and is nonzero exactly when the matrix is invertible.