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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. |
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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. ---- == Diagonalizability == A [[LinearAlgebra/SpecialMatrices#Diagonal_Matrices|diagonal matrix]] has many useful properties. A '''diagonalizable matrix''' is a ''square'' matrix that can be factored into one. Notating the matrix of the [[LinearAlgebra/EigenvaluesAndEigenvectors|eigenvectors]] of '''''A''''' as '''''S''''', a diagonalizable matrix can be factored as '''''A''' = '''SΛS'''^-1^''. '''''Λ''''' will be the diagonal matrix with the [[LinearAlgebra/EigenvaluesAndEigenvectors|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 '''''A'''^2^ = '''SΛ'''^2^'''S'''^-1^'', and more generally '''''A'''^K^ = '''SΛ'''^K^'''S'''^-1^''. A square matrix that is not diagonalizable is called '''defective'''. ---- CategoryRicottone |
Matrix Properties
Matrices can be categorized by whether or not they feature certain properties.
Contents
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.
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 AK = SΛKS-1.
A square matrix that is not diagonalizable is called defective.