6.48.12  Hessenberg matrix reduction: hessenberg SCHUR

A Hessenberg matrix is a square matrix where the coefficients below the sub-principal diagonal are all 0s. The hessenberg command finds a Hessenberg matrix equivalent to a given square matrix.

• hessenberg takes one mandatory argument and one optional argument:
  • A, a matrix.
  • n, an integer, either 0, −1, −2 or a prime number greater than 1 (by default n=0).
• hessenberg(A ⟨ n ⟩) returns a list [P,B] with B=P⁻¹AP and:
  • if n=0, B is a Hessenberg matrix.
  • if n=−1, the calculations are approximate and B is upper triangular.
  • if n=−2, the calculations are approximate, P is orthogonal and B has zero sub-subdiagonal elements.

SCHUR(A) is equivalent to hessenberg(A,-1), which is compatible with HP calculators.

Examples.

• Input:

  A:=[[3,2,2,2,2],[2,1,2,-1,-1],[2,2,1,-1,1],[2,-1,-1,3,1],[2,-1,1,1,2]];

  [P,B]:=hessenberg(A)

  or:
  [P,B]:= hessenberg(A,0);
  Output:

  ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

  1 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 1 1 2 1 4 1 0 1 1 1 0

  ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

  ,

  ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

  3 8 5 5 2 2 2 1 1 2 − 5 4 −1 0 2 1 2 0 0 0 2 3 2 2 0 0 0 13 8 7 2

  ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

  Indeed:
  Input:
  pcar(A)
  and:
  pcar(B)
  both return:
  Output:

  ⎡ ⎣

  1,−10,13,71,−50,−113

  ⎤ ⎦

  and it is easily verified that B=inv(P)*A*P.
• With A as above: Input:
  B:= hessenberg(A,-1);
  Output (to 2 digits):

  ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

  ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

  0.73 −0.057 −0.42 −0.17 −0.51 0.25 −0.53 0.72 −0.38 −0.048 0.35 −0.44 −0.3 0.19 0.74 0.34 0.68 0.17 −0.46 0.43 0.41 0.25 0.44 0.76 −0.063

  ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

  ,

  ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

  6.7 8.7e−15 −2e−13 2.7e−14 −1.4e−13 0.0 4.6 0 0 0 0.0 0.0 −1.9 0 0 0.0 0.0 0.0 1.7 0 0.0 0.0 0.0 −0.0 −1.2

  ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

  ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦