Eigenvalues and eigenvectors of a real matrix.
Computes the eigenvalues and eigenvectors of a matrix A.
If A is diagonalizable, this provides matrices V and D such that A = V*D*V.inv, where D is the diagonal matrix with entries equal to the eigenvalues and V is formed by the eigenvectors.
If A is symmetric, then V is orthogonal and thus A = V*D*V.t
Constructs the eigenvalue decomposition for a square matrix A
# File matrix/eigenvalue_decomposition.rb, line 19 def initialize(a) # @d, @e: Arrays for internal storage of eigenvalues. # @v: Array for internal storage of eigenvectors. # @h: Array for internal storage of nonsymmetric Hessenberg form. raise TypeError, "Expected Matrix but got #{a.class}" unless a.is_a?(Matrix) @size = a.row_count @d = Array.new(@size, 0) @e = Array.new(@size, 0) if (@symmetric = a.symmetric?) @v = a.to_a tridiagonalize diagonalize else @v = Array.new(@size) { Array.new(@size, 0) } @h = a.to_a @ort = Array.new(@size, 0) reduce_to_hessenberg hessenberg_to_real_schur end end
Returns the block diagonal eigenvalue matrix D
# File matrix/eigenvalue_decomposition.rb, line 73 def eigenvalue_matrix Matrix.diagonal(*eigenvalues) end
Returns the eigenvalues in an array
# File matrix/eigenvalue_decomposition.rb, line 59 def eigenvalues values = @d.dup @e.each_with_index{|imag, i| values[i] = Complex(values[i], imag) unless imag == 0} values end
Returns the eigenvector matrix V
# File matrix/eigenvalue_decomposition.rb, line 43 def eigenvector_matrix Matrix.send(:new, build_eigenvectors.transpose) end
Returns the inverse of the eigenvector matrix V
# File matrix/eigenvalue_decomposition.rb, line 50 def eigenvector_matrix_inv r = Matrix.send(:new, build_eigenvectors) r = r.transpose.inverse unless @symmetric r end
Returns an array of the eigenvectors
# File matrix/eigenvalue_decomposition.rb, line 67 def eigenvectors build_eigenvectors.map{|ev| Vector.send(:new, ev)} end
Returns [eigenvector_matrix, #eigenvalue_matrix, #eigenvector_matrix_inv]
# File matrix/eigenvalue_decomposition.rb, line 80 def to_ary [v, d, v_inv] end