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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/MatrixTransposition|transpose]]. {{{ julia> A = [1 2; 2 1] 2×2 Matrix{Int64}: 1 2 2 1 julia> A == A' true }}} ---- == Invertability == A matrix '''''A''''' is '''invertible''' and '''non-singular''' if it can be [[LinearAlgebra/MatrixInversion|inverted]] into matrix '''''A'''^-1''. Not all matrices are invertible. ---- == 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'''''. |
