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Size: 2482
Comment: Diagonalizability 2
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Size: 2157
Comment: Positive definite
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| A [[LinearAlgebra/SpecialMatrices#Diagonal_Matrices|diagonal matrix]] has many useful properties. A '''diagonalizable matrix''' is a ''square'' matrix that can be factored into one. | A [[LinearAlgebra/SpecialMatrices#Diagonal_Matrices|diagonal matrix]] has many useful properties. A '''diagonalizable matrix''' is a ''square'' matrix that can be [[LinearAlgebra/Diagonalization|factored into one]]. |
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| 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'''''. | ---- |
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| This is useful because '''''A'''^2^ = '''SΛ'''^2^'''S'''^-1^'', and more generally '''''A'''^K^ = '''SΛ'''^K^'''S'''^-1^''. | |
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| A square matrix that is not diagonalizable is called '''defective'''. | == Positive Definite == A matrix is '''positive definite''' if it is symmetric and if ''z^T^'''A'''z'' is positive for every vector ''z''. === Positive Semi-definite === A slight modification of the above requirement: a matrix can be called '''positive semi-definite''' if ''z^T^'''A'''z'' is positive ''or zero'' for every vector ''z''. |
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.
Positive Definite
A matrix is positive definite if it is symmetric and if zTAz is positive for every vector z.
Positive Semi-definite
A slight modification of the above requirement: a matrix can be called positive semi-definite if zTAz is positive or zero for every vector z.