WOLFRAM SYSTEM MODELER

# realSchur

Return the real Schur form (rsf) S of a square matrix A, A=QZ*S*QZ' # Wolfram Language

In:= `SystemModel["Modelica.Math.Matrices.realSchur"]`
Out:= # Information

This information is part of the Modelica Standard Library maintained by the Modelica Association.

#### Syntax

```                            S = Matrices.realSchur(A);
(S, QZ, alphaReal, alphaImag) = Matrices.realSchur(A);
```

#### Description

Function realSchur calculates the real Schur form of a real square matrix A, i.e.

``` A = QZ*S*transpose(QZ)
```

with the real nxn matrices S and QZ. S is a block upper triangular matrix with 1x1 and 2x2 blocks in the diagonal. QZ is an orthogonal matrix. The 1x1 blocks contains the real eigenvalues of A. The 2x2 blocks [s11, s12; s21, s11] represents the conjugated complex pairs of eigenvalues, whereas the real parts of the eigenvalues are the elements of the diagonal (s11). The imaginary parts are the positive and negative square roots of the product of the two elements s12 and s21 (imag = +-sqrt(s12*s21)).

The calculation in lapack.dgees is performed stepwise, i.e., using the internal methods of balancing and scaling of dgees.

#### Example

```   Real A[3,3] = [1, 2, 3; 4, 5, 6; 7, 8, 9];
Real T[3,3];
Real Z[3,3];
Real alphaReal;
Real alphaImag;

algorithm
(T, Z, alphaReal, alphaImag):=Modelica.Math.Matrices.realSchur(A);
//   T = [16.12, 4.9,   1.59E-015;
//        0,    -1.12, -1.12E-015;
//        0,     0,    -1.30E-015]
//   Z = [-0.23,  -0.88,   0.41;
//        -0.52,  -0.24,  -0.82;
//        -0.82,   0.4,    0.41]
//alphaReal = {16.12, -1.12, -1.32E-015}
//alphaImag = {0, 0, 0}
```

Math.Matrices.Utilities.reorderRSF

# Syntax

(S, QZ, alphaReal, alphaImag) = realSchur(A)

# Inputs (1)

A Type: Real[:,size(A, 1)] Description: Square matrix

# Outputs (4)

S Type: Real[size(A, 1),size(A, 2)] Description: Real Schur form of A Type: Real[size(A, 1),size(A, 2)] Description: Schur vector Matrix Type: Real[size(A, 1)] Description: Real part of eigenvalue=alphaReal+i*alphaImag Type: Real[size(A, 1)] Description: Imaginary part of eigenvalue=alphaReal+i*alphaImag

# Revisions

• 2010/05/31 by Marcus Baur, DLR-RM