MATLAB Function Reference  rsf2csf

Convert real Schur form to complex Schur form

Syntax

• ````[U,T]` `=` ```rsf2csf(U,T)
``````

Description

The complex Schur form of a matrix is upper triangular with the eigenvalues of the matrix on the diagonal. The real Schur form has the real eigenvalues on the diagonal and the complex eigenvalues in 2-by-2 blocks on the diagonal.

```[U,T] = rsf2csf(U,T) ``` converts the real Schur form to the complex form.

Arguments `U` and `T` represent the unitary and Schur forms of a matrix `A`, respectively, that satisfy the relationships: `A` = `U*T*U'` and `U'*U` = `eye(size(A))`. See `schur` for details.

Examples

Given matrix `A`,

• ``` 1     1     1     3
1     2     1     1
1     1     3     1
-2     1     1     4
```

with the eigenvalues

• ```4.8121    1.9202 + 1.4742i    1.9202 + 1.4742i    1.3474
```

Generating the Schur form of `A` and converting to the complex Schur form

• ```[u,t] = schur(A);
[U,T] = rsf2csf(u,t)
```

yields a triangular matrix `T` whose diagonal (underlined here for readability) consists of the eigenvalues of `A`.

• ```U =

-0.4916       -0.2756 - 0.4411i    0.2133 + 0.5699i   -0.3428
-0.4980       -0.1012 + 0.2163i   -0.1046 + 0.2093i    0.8001
-0.6751        0.1842 + 0.3860i   -0.1867 - 0.3808i   -0.4260
-0.2337        0.2635 - 0.6481i    0.3134 - 0.5448i    0.2466

T =

4.8121        -0.9697 + 1.0778i   -0.5212 + 2.0051i   -1.0067
0         1.9202 + 1.4742i    2.3355              0.1117 + 1.6547i
0              0              1.9202 - 1.4742i    0.8002 + 0.2310i
0              0                   0              1.3474
```

`schur`