rsf2csf

Convert real Schur form to complex Schur form

Syntax

[Unew,Tnew] = rsf2csf(U,T)

Description

[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 $$a\pm i\sqrt{-fe}$$ and $$c\pm i\sqrt{-hg}$$.

• 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

Transform Real Schur Form to Complex Schur Form

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

U — Unitary matrix

matrix

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

T — Schur form

matrix

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

Unew — Transformed unitary matrix

matrix

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

Tnew — Transformed Schur form

matrix

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.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• Code generation does not support sparse matrix inputs for this function.

Thread-Based Environment

Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.

This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

Version History

Introduced before R2006a