= Special Matrices = These '''special matrices''' are core concepts to linear algebra. <> ---- == Identity Matrix == The '''identity matrix''' is a diagonal line of ones in a square matrix of zeros. Any matrix '''''A''''' multiplied by the (appropriately sized) identity matrix returns matrix '''''A'''''. {{{ julia> using LinearAlgebra julia> Matrix{Int8}(I,3,3) 3×3 Matrix{Int8}: 1 0 0 0 1 0 0 0 1 }}} ---- == Permutation Matrices == A '''permutation matrix''' is a square matrix of zeros with a one in each row. Multiplying a permutation matrix by some matrix '''''A''''' ('''''PA''''') results in a row-exchanged '''''A'''''. Multiplying some matrix '''''A''''' by a permutation matrix ('''''AP''''') results in a column-exhanged '''''A'''''. {{{ julia> P = Matrix{Int8}(I,3,3)[:,[3,2,1]] 3×3 Matrix{Int8}: 0 0 1 0 1 0 1 0 0 julia> A = [1 2 3; 4 5 6; 7 8 9] 3×3 Matrix{Int64}: 1 2 3 4 5 6 7 8 9 julia> P * A 3×3 Matrix{Int64}: 7 8 9 4 5 6 1 2 3 julia> A * P 3×3 Matrix{Int64}: 3 2 1 6 5 4 9 8 7 }}} The transpose permutation matrix is the same as the inverse permutation matrix: '''''P'''^T^ = '''P'''^-1^''. The transpose permutation matrix multiplied by the permutation matrix is the same as the identity matrix: '''''P'''^T^'''P''' = '''I''''' === Counting Permutations === For 3 by 3 matrices, there are 6 possible permutation matrices. They are often denoted based on the rows they exchange, such as '''''P''',,2 3,,''. {{{ ┌ ┐ ┌ ┐ ┌ ┐ ┌ ┐ ┌ ┐ ┌ ┐ │ 1 0 0│ │ 1 0 0│ │ 0 1 0│ │ 0 1 0│ │ 0 0 1│ │ 0 0 1│ │ 0 1 0│ │ 0 0 1│ │ 1 0 0│ │ 0 0 1│ │ 1 0 0│ │ 0 1 0│ │ 0 0 1│ │ 0 1 0│ │ 0 0 1│ │ 1 0 0│ │ 0 1 0│ │ 1 0 0│ └ ┘ └ ┘ └ ┘ └ ┘ └ ┘ └ ┘ (identity matrix) P P (and so on...) 2,3 1,2 }}} For any ''n'' by ''n'' matrix, there are ''n''! possible permutation matrices. ---- == Upper Triangular Matrices == If a square matrix has only zeros below the diagonal, it is an '''upper triangular matrix'''. [[LinearAlgebra/Elimination|Gauss-Jordan elimination]] results in a '''row echelon form''' of '''''A''''' which, if '''''A''''' is square, is also upper triangular. [[LinearAlgebra/Orthonormalization|Gram-Schmidt orthonormalization]] is characterized as '''''A = QR''''' where '''''R''''' is upper triangular. ---- == Lower Triangular Matrices == If a square matrix has only zeros above the diagonal, it is a '''lower triangular matrix'''. If [[LinearAlgebra/Elimination|Gauss-Jordan elimination]] is continued into backwards elimination, it results in a '''reduced row echelon form''' of '''''A'''''. If '''''A''''' is square, it will be both upper and lower triangular. ---- == Diagonal Matrices == A '''diagonal matrix''' is a diagonal line of numbers in a square matrix of zeros. The columns of a diagonal matrix are its [[LinearAlgebra/EigenvaluesAndEigenvectors|eigenvectors]], and the numbers in the diagonal are the [[LinearAlgebra/EigenvaluesAndEigenvectors|eigenvalues]]. ---- CategoryRicottone