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schur

Schur decomposition

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    Syntax

    T = schur(A)
    T = schur(A,mode)
    [U,T] = schur(___)

    Description

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

    example

    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.

    example

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

    example

    Examples

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    Open Live Script

    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
    Open Live Script

    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
    Open Live Script

    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 = 
6.7711e-15

    norm(U'*U - eye(size(U)))
    ans = 
5.0471e-16

    Input Arguments

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

    Data Types: single | double
    Complex Number Support: Yes

    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

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    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.

    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

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    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.

    6-by-6 upper quasitriangular matrix 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=UU*=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

    Version History

    Introduced before R2006a

    See Also

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