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---- == Symmetry == A '''symmetric matrix''' is equal to its [[LinearAlgebra/MatrixTransposition|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 [[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. |
Only a square matrix can be idempotent. |
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| == Diagonalizability == | == 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 '''positive definite matrix''' is a symmetric matrix where all [[LinearAlgebra/EigenvaluesAndEigenvectors|eigenvalues]] are positive. Following from the properties of all symmetric matrices, all pivots are also positive. Necessarily the [[LinearAlgebra/Determinants|determinant]] is also positive, and all subdeterminants are also positive. |
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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 Semi-definite === A slight modification of the above requirement: 0 is also allowable. |
Matrix Properties
Matrices can be categorized by whether or not they feature certain properties.
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.
Only a square matrix can be idempotent.
Positive Definite
A positive definite matrix is a symmetric matrix where all eigenvalues are positive. Following from the properties of all symmetric matrices, all pivots are also positive. Necessarily the determinant is also positive, and all subdeterminants are also positive.
Positive Semi-definite
A slight modification of the above requirement: 0 is also allowable.