# rsf2csf

Convert real Schur form to complex Schur form

## Description

example

[Unew,Tnew] = rsf2csf(U,T) transforms the outputs of [U,T] = schur(X) for real matrices X from real Schur form to complex Schur form. This operation transforms how the eigenvalues of X are expressed in T, and transforms U such that X = Unew*Tnew*Unew' and Unew'*Unew = eye(size(X)).

• In real Schur form, T has real eigenvalues on the diagonal, and complex eigenvalues are expressed as 2-by-2 real blocks along the main diagonal:

$\left[\begin{array}{cccccc}{\lambda }_{1}& {t}_{12}& {t}_{13}& {t}_{14}& {t}_{15}& \cdots \\ & a& e& {t}_{24}& {t}_{25}& \cdots \\ & f& a& {t}_{34}& {t}_{35}& \cdots \\ & & & c& g& \cdots \\ & & & h& c& \cdots \\ & & & & & \ddots \end{array}\right]$

The eigenvalues represented by these blocks are $a±i\sqrt{-fe}$ and $c±i\sqrt{-hg}$.

• In complex Schur form, Tnew is upper triangular with all eigenvalues, real or complex, on the main diagonal:

$\left[\begin{array}{cccccc}{\lambda }_{1}& tne{w}_{12}& tne{w}_{13}& tne{w}_{14}& tne{w}_{15}& \cdots \\ & a+bi& tne{w}_{23}& tne{w}_{24}& tne{w}_{25}& \cdots \\ & & a-bi& tne{w}_{34}& tne{w}_{35}& \cdots \\ & & & c+di& tne{w}_{45}& \cdots \\ & & & & c-di& \cdots \\ & & & & & \ddots \end{array}\right]$

## Examples

collapse all

Apply Schur decomposition to a real matrix, and then transform the matrix factors so that the eigenvalues are directly on the main diagonal.

Create a real matrix and calculate the Schur decomposition. The U factor is unitary so that ${\mathit{U}}^{\mathit{T}}\mathit{U}={\mathit{I}}_{\mathit{N}}$, and the T factor is in real Schur form with complex conjugate eigenvalue pairs expressed as 2-by-2 blocks on the diagonal.

X = [1     1     1     3
1     2     1     1
1     1     3     1
-2     1     1     4];
[U,T] = schur(X)
U = 4×4

-0.4916   -0.4900   -0.6331   -0.3428
-0.4980    0.2403   -0.2325    0.8001
-0.6751    0.4288    0.4230   -0.4260
-0.2337   -0.7200    0.6052    0.2466

T = 4×4

4.8121    1.1972   -2.2273   -1.0067
0    1.9202   -3.0485   -1.8381
0    0.7129    1.9202    0.2566
0         0         0    1.3474

T has two real eigenvalues on the diagonal and one 2-by-2 block representing a complex conjugate pair of eigenvalues.

Transform U and T so that Tnew is upper triangular with the eigenvalues on the diagonal, and Unew satisfies X = Unew*Tnew*Unew'.

[Unew,Tnew] = rsf2csf(U,T)
Unew = 4×4 complex

-0.4916 + 0.0000i  -0.2756 - 0.4411i   0.2133 + 0.5699i  -0.3428 + 0.0000i
-0.4980 + 0.0000i  -0.1012 + 0.2163i  -0.1046 + 0.2093i   0.8001 + 0.0000i
-0.6751 + 0.0000i   0.1842 + 0.3860i  -0.1867 - 0.3808i  -0.4260 + 0.0000i
-0.2337 + 0.0000i   0.2635 - 0.6481i   0.3134 - 0.5448i   0.2466 + 0.0000i

Tnew = 4×4 complex

4.8121 + 0.0000i  -0.9697 + 1.0778i  -0.5212 + 2.0051i  -1.0067 + 0.0000i
0.0000 + 0.0000i   1.9202 + 1.4742i   2.3355 - 0.0000i   0.1117 + 1.6547i
0.0000 + 0.0000i   0.0000 + 0.0000i   1.9202 - 1.4742i   0.8002 + 0.2310i
0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   1.3474 + 0.0000i

## Input Arguments

collapse all

Unitary matrix, specified as the matrix returned by [U,T] = schur(X). The matrix U satisfies U'*U = eye(size(X)).

Data Types: single | double
Complex Number Support: Yes

Schur form, specified as the matrix returned by [U,T] = schur(X). The matrix T satisfies X = U*T*U'. The Schur form has real eigenvalues on the diagonal, and complex eigenvalues are expressed as 2-by-2 real blocks along the main diagonal.

Data Types: single | double
Complex Number Support: Yes

## Output Arguments

collapse all

Transformed unitary matrix, returned as a matrix. The matrix Unew satisfies Unew'*Unew = eye(size(X)).

Transformed Schur form, returned as a matrix. Tnew is upper triangular with the eigenvalues of X on the diagonal, and it satisfies X = Unew*Tnew*Unew'.

## Tips

• You can use ordeig to obtain the same eigenvalue ordering as rsf2csf from the results of a Schur decomposition. However, rsf2csf also returns the remainder of the Schur matrix T and Schur vector matrix U, transformed to complex representation.