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

Methods
D
E
N
T
V
Class Public methods
new(a)

Constructs the eigenvalue decomposition for a square matrix A

# File lib/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
Instance Public methods
d()
Alias for: eigenvalue_matrix
eigenvalue_matrix()

Returns the block diagonal eigenvalue matrix D

Also aliased as: d
# File lib/matrix/eigenvalue_decomposition.rb, line 73
def eigenvalue_matrix
  Matrix.diagonal(*eigenvalues)
end
eigenvalues()

Returns the eigenvalues in an array

# File lib/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
eigenvector_matrix()

Returns the eigenvector matrix V

Also aliased as: v
# File lib/matrix/eigenvalue_decomposition.rb, line 43
def eigenvector_matrix
  Matrix.send(:new, build_eigenvectors.transpose)
end
eigenvector_matrix_inv()

Returns the inverse of the eigenvector matrix V

Also aliased as: v_inv
# File lib/matrix/eigenvalue_decomposition.rb, line 50
def eigenvector_matrix_inv
  r = Matrix.send(:new, build_eigenvectors)
  r = r.transpose.inverse unless @symmetric
  r
end
eigenvectors()

Returns an array of the eigenvectors

# File lib/matrix/eigenvalue_decomposition.rb, line 67
def eigenvectors
  build_eigenvectors.map{|ev| Vector.send(:new, ev)}
end
to_a()
Alias for: to_ary
to_ary()

Returns [eigenvector_matrix, #eigenvalue_matrix, #eigenvector_matrix_inv]

Also aliased as: to_a
# File lib/matrix/eigenvalue_decomposition.rb, line 80
def to_ary
  [v, d, v_inv]
end
v()
Alias for: eigenvector_matrix
v_inv()