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For a symmetric matrix, the [[LinearAlgebra/EigenvaluesAndEigenvectors|eigenvalues]] are always real and the [[LinearAlgebra/EigenvaluesAndEigenvectors|eigenvectors]] can be written as perpendicular vectors. This means that [[LinearAlgebra/Diagonalization|diagonalization]] of a symmetric matrix is expressed as '''''A''' = '''QΛQ'''^-1^ = '''QΛQ'''^T^'', by using the [[LinearAlgebra/Orthogonality#Matrices|orthonormal eigenvectors]]. Symmetric matrices are combinations of perpendicular [[LinearAlgebra/Projections|projection matrices]]. For a symmetric matrix, the signs of the pivots are the same as the signs of the eigenvalues. |
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| A matrix is '''positive definite''' if it is symmetric and if ''z^T^'''A'''z'' is positive for every vector ''z''. | 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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| 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''. | 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.
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'
trueFor a symmetric matrix, the eigenvalues are always real and the eigenvectors can be written as perpendicular vectors. This means that diagonalization of a symmetric matrix is expressed as A = QΛQ-1 = QΛQT, by using the orthonormal eigenvectors.
Symmetric matrices are combinations of perpendicular projection matrices.
For a symmetric matrix, the signs of the pivots are the same as the signs of the eigenvalues.
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 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.