schur
Syntax
schur(obj, [sort])
Arguments
obj is a square matrix.
sort is a string. It is used to reorder the factors according to a specified ordering of the eigenvalues. The value can be ‘lhp’ (eigenvalue is a negative real number), ‘rhp’ (eigenvalue is a positive real number), ‘iuc’ (the absolute value of a complex eigenvalue<=1.0), ‘ouc’ (the absolute value of a complex eigenvalue>1.0).
Details
Compute the Schur decomposition of a square matrix.
Suppose the input is the square matrix A:
If sort is not specified, return 2 matrices: T (Schur form of A, an upper triangular matrix) and an unitary matrix Z (the transpose matrix of Z is equal to its inverse matrix), so that A = Z*T*Z-1.
If sort is specified, the function will also return an integer indicating the number of eigenvalues that meet the sorting conditions.
Examples
$ m=matrix([[0,0,1],[2,1,0],[2,2,1]]);
$ T,Z=schur(m)
$ T;
#0 |
#1 |
#2 |
|---|---|---|
2.658967 |
1.424405 |
-1.929334 |
0 |
-0.329484 |
-0.490637 |
0 |
1.311789 |
-0.329484 |
$ Z
#0 |
#1 |
#2 |
|---|---|---|
0.727116 |
-0.601562 |
0.330796 |
0.528394 |
0.798019 |
0.289768 |
0.438294 |
0.035904 |
-0.898114 |
$ T,Z,s=schur(m, 'lhp');
$ T;
#0 |
#1 |
#2 |
|---|---|---|
-0.329484 |
1.570974 |
2.251318 |
-0.40969 |
-0.329484 |
-0.092398 |
0 |
0 |
2.658967 |
$ Z
#0 |
#1 |
#2 |
|---|---|---|
0.703818 |
-0.632169 |
0.324042 |
0.509043 |
0.766983 |
0.390655 |
-0.495495 |
-0.109999 |
0.861618 |
$ s
2
$ T,Z,s=schur(m, 'rhp');
$ s;
1
$ m=matrix([[0,0,9],[-2,1,0],[2,2,1]]);
$ T,Z,s=schur(m, 'iuc');
$ s;
0
$ T,Z,s=schur(m, 'ouc');
$ s;
1