schur

Schur decomposition

Syntax

T = schur(A)

T = schur(A,mode)

[U,T] = schur(___)

Description

T = schur(A) returns the Schur matrix of A.

T = schur(A,mode), if A is a real matrix, returns a real quasitriangular Schur matrix if mode is "real" or returns a complex triangular Schur matrix if mode is "complex". If A is a complex matrix, then schur returns the complex Schur form regardless of the value of mode.

[U,T] = schur(___) also returns a unitary matrix U such that A = U*T*U'.

Examples

Schur Form of Matrix

Create a 3-by-3 matrix and calculate its Schur form. The result is a matrix with the eigenvalues of A (which are 1, 2, and 3) on the diagonal.

A = [-149 -50 -154; 537 180 546; -27 -9 -25];
T = schur(A)

T = 3×3

    1.0000   -7.1119  815.8706
         0    2.0000   55.0236
         0         0    3.0000

The large off-diagonal elements indicate that this matrix has poorly conditioned eigenvalues, which can be confirmed by finding the condition numbers using the condeig function.

condeig(A)

ans = 3×1

  603.6390
  395.2366
  219.2920

Schur Form of Matrix with Complex Eigenvalues

Create a 3-by-3 matrix with complex eigenvalues.

A = [3 1 1; 0 2 0; -2 1 1]

A = 3×3

     3     1     1
     0     2     0
    -2     1     1

Calculate the real Schur form of A. The 2-by-2 block matrix on the diagonal represents a pair of conjugate complex eigenvalues.

T1 = schur(A)

T1 = 3×3

    2.0000    0.3820    1.3764
   -2.6180    2.0000    0.3249
         0         0    2.0000

Find the eigenvalues of the real Schur form using the ordeig function.

ordeig(T1)

ans = 3×1 complex

   2.0000 + 1.0000i
   2.0000 - 1.0000i
   2.0000 + 0.0000i

Calculate the complex Schur form of A by specifying mode as "complex". The diagonal values of T2 are the eigenvalues of the Schur form of A, which are the same as the output of ordeig.

T2 = schur(A,"complex")

T2 = 3×3 complex

   2.0000 + 1.0000i   2.2361 + 0.0000i  -0.3035 - 0.4911i
   0.0000 + 0.0000i   2.0000 - 1.0000i   1.2858 + 0.1159i
   0.0000 + 0.0000i   0.0000 + 0.0000i   2.0000 + 0.0000i

Schur Decomposition of Matrix

Create a 3-by-3 magic square matrix and calculate its Schur decomposition factors.

A = magic(3);
[U,T] = schur(A)

U = 3×3

   -0.5774   -0.8131   -0.0749
   -0.5774    0.4714   -0.6667
   -0.5774    0.3416    0.7416

T = 3×3

   15.0000    0.0000   -0.0000
         0    4.8990   -3.4641
         0         0   -4.8990

Verify that the norms of A-U*T*U' and U'*U - eye(size(U)) are 0, within machine precision.

norm(A-U*T*U')

ans = 
1.0954e-14

norm(U'*U - eye(size(U)))

ans = 
9.2217e-16

Input Arguments

A — Input matrix

square matrix

Input matrix, specified as a real or complex square matrix. A must be dense.

Data Types: single | double
Complex Number Support: Yes

mode — Schur form system

"real" (default) | "complex"

Schur form system, specified as "real" or "complex".

• If A is real, then mode can be "real" or "complex". If mode is "real", then schur returns real quasitriangular T. If mode is "complex", then schur returns complex triangular T.

• If A is complex, then schur ignores mode and returns the complex Schur form, which is upper triangular with the eigenvalues of A on the diagonal.

Output Arguments

U — Schur vectors

unitary matrix

Schur vectors, returned as a unitary matrix that satisfies A = U*T*U'. Each column of U corresponds to a Schur vector. If the Schur matrix T is triangular, then the first column of U is an eigenvector of A corresponding to the first element of T.

T — Schur matrix

square matrix

Schur matrix, returned as a real or complex square matrix. If mode is "complex", then T is triangular. If mode is "real", then T is quasitriangular.

More About

Quasitriangular Matrix

An upper quasitriangular matrix can result from the Schur decomposition or generalized Schur (QZ) decomposition of a real matrix. An upper quasitriangular matrix is block upper triangular, with 1-by-1 and 2-by-2 blocks of nonzero values along the diagonal.

The eigenvalues of these diagonal blocks are also the eigenvalues of the matrix. The 1-by-1 blocks correspond to real eigenvalues, and the 2-by-2 blocks correspond to complex conjugate eigenvalue pairs.

Unitary Matrix

An invertible complex square matrix U is unitary if its conjugate transpose is also its inverse, that is, if $$U^{*}U=U{U}^{*}=I$$.

Tips

• You can use the rsf2csf function to convert the real Schur form of a matrix to its complex Schur form. [U,T] = rsf2csf(A) returns the same outputs as [U,T] = schur(A,"complex").

Extended Capabilities

C/C++ Code Generation

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

Usage notes and limitations:

• Generated code can return a different Schur decomposition than MATLAB^® returns.

• When the input matrix contains a nonfinite value, the generated code does not issue an error. Instead, the output contains NaN values.

• 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