schur#

Purpose#

Computes the real or complex Schur form of a square matrix with the option to sort the eigenvalues.

Format#

{ S, Z } = schur(A[, flag[, sort_type]])#

Parameters:

A (KxK matrix) – data
flag (string) –
Optional input, to control whether output should be in real or complex Schur form. Valid options include:
- "complex"
- "real"

sort_type (string) –

Optional input, specifying how to sort the eigenvalues. Options include:

1	”udi”	Absolute value of the eigenvalue less than 1.0. (Unit disk inside)
2	”udo”	Absolute value of the eigenvalue greater than or equal to 1.0. (Unit disk outside)
3	”lhp”	Value of the real portion of the eigenvalue less than 0. (Left hand plane)
4	”rhp”	Value of the real portion of the eigenvalue greater than 0. (Right hand plane)
5	”ref”	Real eigenvalues first. (Complex portion less than imagtol see remarks section)
6	”cef”	Complex eigenvalues first. (Complex portion greater than imagtol see remarks section)

Returns:

S (KxK matrix) – Schur form.
Z (KxK matrix) – transformation matrix.

Examples#

Real matrix with all real eigenvalues#

// Create a 2 x 2 matrix
A = { 7 -2,
     12 -5 };

// Calculate eigenvalues of 'A'
lambda = eig(A);

After the code above, lambda should equal:

 4.4641
-2.4641

// Continuing with 'A' from above
{ S, Z } = schur(A);

Now S and Z should equal:

S = 4.4641   -14.000  Z =  0.6193   -0.7852
         0   -2.4641       0.7852    0.6193

Real matrix with some complex eigenvalues#

// Create a 3 x 3 matrix
A = {  1  -4  -1,
       3  -1   9,
      -9   1  -2 };

// Calculate real schur form, with complex eigenvalues
// stored as 2 x 2 blocks on the diagonal
{ S_r, Z_r } = schur(A, "real");

// Calculate complex schur form
{ S_c, Z_c } = schur(A, "complex");

After the code above:

S_r = -7.2208   8.6875   -1.7726
      -4.4007  -1.1773    1.5989
            0        0    6.3981

Z_r = -0.1304  0.7105 -0.6915
       0.6913  0.5651  0.4502
      -0.7107  0.4193  0.5649

S_c = -4.1991+5.3945i  -2.5084+6.9720i   1.0168-0.7763i
                 0+0i  -4.1991-5.3945i   1.9825-0.3630i
                 0+0i             0+0i        6.3981+0i

Z_c =  0.3275 -   0.1759i   0.5326 +   0.3160i  -0.6915
       0.1102 -   0.7700i   0.0023 +   0.4385i   0.4502
       0.3132 +   0.3984i   0.6502 +   0.0373i   0.5649

Complex matrix with sorted eigenvalues#

// Create a 3 x 3 complex matrix
A = { -1.9615 + 0.4382i   0.0655 + 0.6913i  -1.1424 + 0.1997i,
       0.1244 + 0.3783i  -0.2821 + 0.0588i   0.4854 + 0.4700i,
       1.1271 + 0.7045i  -1.5245 - 0.9966i   1.4969 + 0.4450i };

// Place eigenvalues in unit circle at top-left
{ S, Z } = schur(A, "complex", "udi");

After the code above:

     -0.3548 +   0.8005i   2.4873 -   0.4942i  -1.3144 +   0.7286i
S =   0.0000               1.0504 -   0.5581i   0.1763 +   0.7846i
      0.0000               0.0000              -1.4423 +   0.6996i

      0.3692 -   0.2393i  -0.0144 +   0.2838i   0.5890 -   0.6155i
Z =  -0.3907 +   0.2625i   0.5407 -   0.5251i   0.4161 -   0.1930i
     -0.7530 -   0.1336i  -0.5813 +   0.1154i   0.2225 -   0.1201i

Remarks#

If a real matrix is passed in without a flag variable, the real Schur form will be returned. If a complex matrix is passed in without a flag variable, GAUSS will check to see if any of the imaginary elements are greater than imagtol (2.23e-16 by default). If any imaginary elements are greater than imagtol, the complex Schur form will be calculated, otherwise the real Schur form will be returned. If a real flag is passed in with a complex matrix, the flag will be ignored and the complex Schur factorization will be returned.

The real Schur form is an upper quasi-triangular matrix, that is, it is block triangular where the blocks are 2x2 submatrices which correspond to complex eigenvalues of A. If A has no complex eigenvalues, S will be strictly upper triangular. To convert the real Schur form of S to the complex Schur form, use the Run-Time Library function schtoc().

Z is an orthogonal matrix that transforms A into S and vice versa. Thus

S = Z'*A*Z;

and since Z is orthogonal,

A = Z*S*Z';