Linear Algebra
Cayley–Hamilton Theorem
Every square matrix satisfies its own characteristic polynomial
Spectral Theorem
Every real symmetric matrix is orthogonally diagonalizable
Rank–Nullity Theorem
The dimensions of image and kernel of a linear map sum to the domain dimension
Determinant Multiplicativity
The determinant of a product equals the product of determinants
Cramer's Rule
Explicit determinant formula for the solution of a linear system
Lagrange Interpolation
Given n distinct nodes, there is a unique polynomial of degree < n passing through all n points.
Newton's Identities
Power sums and elementary symmetric polynomials satisfy a mutual recursion that lets each determine the other.
LU Decomposition
Every invertible matrix factors as a unit-lower-triangular matrix times an upper-triangular matrix.
QR Decomposition
Every invertible real matrix factors as a product of an orthogonal matrix and an upper-triangular matrix.
Singular Value Decomposition
Every real matrix factors as U times a diagonal matrix of singular values times V-transpose.
Gram–Schmidt Orthogonalization
Any linearly independent sequence can be transformed into an orthogonal sequence spanning the same subspace.
Hadamard's Inequality
The absolute value of a determinant is at most the product of the Euclidean norms of its columns.
Cauchy–Binet Formula
The determinant of a product AB equals the sum over all maximal minors of A times the corresponding minors of B.
Jordan Canonical Form
Every square matrix over an algebraically closed field is similar to a direct sum of Jordan blocks.
Schur Decomposition
Every square complex matrix is unitarily similar to an upper triangular matrix.
Polar Decomposition
Every invertible matrix factors as a unitary times a positive definite matrix.
Sylvester Determinant Theorem
det(Im + AB) = det(In + BA) for any m×n matrix A and n×m matrix B.
Trace Cyclic Property
The trace of a product of matrices is invariant under cyclic permutations: tr(AB) = tr(BA).
Minimal Polynomial
The minimal polynomial of a matrix is the monic polynomial of least degree that annihilates it.
Diagonalizability Criterion
A matrix is diagonalizable if and only if its minimal polynomial is squarefree.
Companion Matrix
The companion matrix of a monic polynomial has that polynomial as its characteristic polynomial.