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---- == Orthonormality == A [[LinearAlgebra/Orthogonality#Matrices|matrix with orthonormal columns]] has several important properties. A matrix '''''A''''' can be [[LinearAlgebra/Orthonormalization|orthonormalized]] into '''''Q'''''. === Orthogonality === An '''orthogonal matrix''' is a ''square'' matrix with orthonormal columns. ---- CategoryRicottone |
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 A is invertible and non-singular if it can be inverted into matrix A^-1. Not all matrices are invertible.
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.