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| Clearly only a square matrix can be symmetric. 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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| == Invertability == | == Invertible == |
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| A matrix is '''invertible''' and '''non-singular''' if the [[LinearAlgebra/Determinants|determinant]] is non-zero. | A matrix is '''invertible''' if the [[LinearAlgebra/Determinants|determinant]] is not zero. A matrix that is invertible is also called '''non-singular''' and '''non-degenerate'''. A matrix that is non-invertible is also called '''singular''' and '''degenerate'''. It has a determinant of zero. Only a square matrix can be invertible. Also, an invertible matrix can be [[LinearAlgebra/MatrixInversion|inverted]]. Invertibility does not determine the existance of ''any'' inverses though. Chiefly, a non-square matrix (which by definition cannot be invertible) may still have a right or left inverse. |
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| Only a square matrix can be idempotent. |
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| == Orthonormality == | == Orthogonality == |
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| A [[LinearAlgebra/Orthogonality#Matrices|matrix with orthonormal columns]] has several important properties. A matrix '''''A''''' can be [[LinearAlgebra/Orthonormalization|orthonormalized]] into '''''Q'''''. | A square matrix with [[LinearAlgebra/Orthogonality#Matrices|orthonormal columns]] is called '''orthogonal'''. |
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| Some matrices can be [[LinearAlgebra/Orthonormalization|orthonormalized]]. They must be invertible at minimum. | |
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| Orthogonal matrices have several properties: | |
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| === Orthogonality === An '''orthogonal matrix''' is a ''square'' matrix with orthonormal columns. |
* '''''Q'''^T^'''Q''' = '''QQ'''^T^ = '''I'''''. * '''''Q'''^T^ = '''Q'''^-1^'' * ''|'''Q'''| = 1'' or ''-1'' always |
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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'
trueClearly only a square matrix can be symmetric.
For 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.
Invertible
A matrix is invertible if the determinant is not zero. A matrix that is invertible is also called non-singular and non-degenerate.
A matrix that is non-invertible is also called singular and degenerate. It has a determinant of zero.
Only a square matrix can be invertible.
Also, an invertible matrix can be inverted. Invertibility does not determine the existance of any inverses though. Chiefly, a non-square matrix (which by definition cannot be invertible) may still have a right or left inverse.
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.
Orthogonality
A square matrix with orthonormal columns is called orthogonal.
Some matrices can be orthonormalized. They must be invertible at minimum.
Orthogonal matrices have several properties:
QTQ = QQT = I.
QT = Q-1
|Q| = 1 or -1 always
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.