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A matrix '''''A''''' is '''invertible''' and '''non-singular''' if it can be [[LinearAlgebra/MatrixInversion|inverted]] into matrix '''''A'''^-1''. Not all matrices are invertible. A matrix is '''invertible''' and '''non-singular''' if the [[LinearAlgebra/Determinants|determinant]] is non-zero.

Matrix Properties

Matrices can be categorized by whether or not they feature certain properties.

Contents

  1. Matrix Properties
    1. Symmetry
    2. Invertability
    3. Idempotency
    4. Orthonormality
      1. Orthogonality

Symmetry

A symmetric matrix is equal to its transpose. 

julia> A = [1 2; 2 1]
2×2 Matrix{Int64}:
 1  2
 2  1

julia> A == A'
true

Invertability

A matrix is invertible and non-singular if the determinant is non-zero.

Idempotency

An idempotent matrix can be multiplied by some matrix A any number of times and the first product will continue to be returned. In other words, A2 = A.

For example, the projection matrix P is characterized as H(HTH)-1HT. If this were squared to H(HTH)-1HTH(HTH)-1HT, then per the core principle of inversion (i.e., AA-1 = I), half of the terms would cancel out. P2 = P.

Orthonormality

A matrix with orthonormal columns has several important properties. A matrix A can be orthonormalized into Q.

Orthogonality

An orthogonal matrix is a square matrix with orthonormal columns.

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