Linear Algebra

For a linear map $T: V \to W$ between finite-dimensional vector spaces, $\dim(V) = \text{rank}(T) + \dim(\ker(T))$.

Every square matrix over a commutative ring satisfies its own characteristic equation; that is, if $p(\lambda) = \det(\lambda I - A)$ is the characteristic polynomial of $A$, then $p(A) = 0$.

Any symmetric matrix with real entries can be diagonalized by an orthogonal matrix, and its eigenvalues are all real.

A real square matrix with positive entries has a unique largest real eigenvalue and a corresponding eigenvector with strictly positive components.

The number of positive, negative, and zero coefficients in the diagonal form of a real quadratic form is invariant under change of basis.

For any vectors $u$ and $v$ in an inner product space, $|\langle u, v \rangle|^2 \le \langle u, u \rangle \cdot \langle v, v \rangle$.

Any complex matrix $M$ can be factored as $U\Sigma V^*$, where $U$ and $V$ are unitary matrices and $\Sigma$ is a diagonal matrix with non-negative real numbers.

Every eigenvalue of a square matrix lies within at least one of the Gershgorin discs defined by the diagonal entries and row sums.

Gives a variational characterization of eigenvalues of Hermitian matrices.

Any square matrix over an algebraically closed field is similar to a block diagonal matrix of Jordan blocks.

A Hermitian, positive-definite matrix can be decomposed into the product of a lower triangular matrix and its conjugate transpose.

Any quadratic form can be transformed into a sum of squares by an orthogonal change of variables.

Any matrix can be factored into the product of a unitary matrix and a positive-semidefinite Hermitian matrix.

Provides variational characterization of eigenvalues of compact Hermitian operators.

Existence of SVD for any matrix.

Analogue of reduced echelon form for matrices over integers.

Explicit formula for the solution of a system of linear equations using determinants.

Diagonal form for matrices over a principal ideal domain.

Low-rank approximation of matrices.

Spectral radius formula.

Bound on the sensitivity of eigenvalues.