Identity

A matrix with diagonal elements 1 and off-diagonal elements 0

Determinant

Determinants are only defined for square matrices. Let A be an n by n matrix with elements {a_ij}. The matrix M_ij here is a subset of A known as the minor, formed by eliminating row i and column j from A. An n by n matrix, A, with Det=0 is described as singular, and such a matrix has no inverse. If Det(A) is very close to 0 it is described as ill-conditioned

|A|, Det(A)

Inverse

The matrix equivalent of division in conventional algebra. For a matrix, A, to be invertible its determinant must be non-zero, and ideally not very close to zero. A matrix that has an inverse is by definition non-singular. A symmetric real-valued matrix is positive definite if all its eigenvalues are positive, whereas a positive semi-definite matrix allows for some eigenvalues to be 0. A matrix, A, that is invertible satisfies the relation AA^‑1=I

A^‑1

Transpose

A matrix operation in which the rows and columns are transposed, i.e. in which elements a_ij are swapped with a_ji for all i,j. The inverse of a transposed matrix is the same as the transpose of the matrix inverse

A^T or A^¢

(A^T)^–1=(A^‑1)^T

Symmetric

A matrix in which element a_ij=a_ji for all i,j

A=A^T

Trace

The sum of the diagonal elements of a matrix, a_ii — the sum of the eigenvalues of a matrix equals its trace

Tr(A)

Eigenvalue, Eigenvector

If A is a real-valued k by k square matrix and x is a non-zero real-valued vector, then a scalar λ that satisfies the equation shown in the adjacent column is known as an eigenvalue of A and x is an eigenvector of A. There are k eigenvalues of A, each with a corresponding eigenvector. The matrix A can be decomposed into three parts, as shown, where E is a matrix of its eigenvectors and D is a diagonal matrix of its eigenvalues

(A‑λI)x=0

A=EDE^‑1 (diagonalization)