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| = Matrix Properties = Matrices can be categorized by whether or not they feature certain '''properties'''. <<TableOfContents>> ---- == Symmetry == A '''symmetric matrix''' is equal to its [[LinearAlgebra/Transposition|transpose]]. {{{ julia> A = [1 2; 2 1] 2×2 Matrix{Int64}: 1 2 2 1 julia> A == A' true }}} 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/Projection|projection matrices]]. For a symmetric matrix, the signs of the pivots are the same as the signs of the eigenvalues. ---- == 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, '''''A'''^2^ = '''A'''''. For example, the [[LinearAlgebra/Projections|projection matrix]] '''''P''''' is characterized as '''H'''('''H'''^T^'''H''')^-1^'''H'''^T^. If this were squared to '''H'''('''H'''^T^'''H''')^-1^'''H'''^T^'''H'''('''H'''^T^'''H''')^-1^'''H'''^T^, then per the core principle of [[LinearAlgebra/MatrixInversion|inversion]] (i.e., '''''AA'''^-1^ = '''I'''''), half of the terms would cancel out. '''''P'''^2^ = '''P'''''. Only a square matrix can be idempotent. ---- == Orthogonality == A square matrix with [[LinearAlgebra/Orthogonality#Matrices|orthonormal columns]] is called '''orthogonal'''. Some matrices can be [[LinearAlgebra/Orthonormalization|orthonormalized]]. They must be invertible at minimum. Orthogonal matrices have several properties: * '''''Q'''^T^'''Q''' = '''QQ'''^T^ = '''I'''''. * '''''Q'''^T^ = '''Q'''^-1^'' * ''|'''Q'''| = 1'' or ''-1'' always ---- == Diagonalizability == 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]]. ---- == Positive Definite == 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. === Positive Semi-definite === A slight modification of the above requirement: 0 is also allowable. ---- CategoryRicottone |
