Matrix Properties

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

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(H^TH)^-1H^T. If this were squared to H(H^TH)^-1H^TH(H^TH)^-1H^T, 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.

Diagonalizability

A diagonal matrix has many useful properties. A diagonalizable matrix is a square matrix that can be factored into one.

Notating the matrix of the eigenvectors of A as S, a diagonalizable matrix can be factored as A = SΛS^-1. Λ will be the diagonal matrix with the eigenvalues of A in the diagonal. In other words, A can be rewritten as a eigennormalized (i.e. transformed by S) then un-eigennormalized (i.e. transformed by S^-1) diagonal matrix Λ.

This is useful because A² = SΛ²S^-1, and more generally A^K = SΛ^KS^-1.

A square matrix that is not diagonalizable is called defective.

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