Schur decomposition

As Schur decomposition or Schur normal form ( according to Isay Schur ) is called in linear algebra, a branch of mathematics, an important matrix decomposition, specifically a Trigonalisierungsverfahren.

Definition

Is a square matrix with entries from (ie, either for or stands for ). Decomposes the characteristic polynomial of over into linear factors, then there exists a unitary matrix, so that

An upper triangular matrix.

Comments

  • There is an upper triangular matrix, it can be represented as the sum of a diagonal matrix and a strict upper triangular matrix ():
  • Is unique up to the order of the diagonal elements and is referred to as the diagonal portion of the Schur decomposition.
  • Is nilpotent, generally only unique with respect to its Frobenius norm and is called the nilpotent share the Schur decomposition.
  • The Frobenius norm of exactly 0 if is normal.
  • Because of the similarity of the output matrix and the upper triangular matrix sitting in the diagonal of the eigenvalues ​​of.
  • Is a regular matrix, then even a diagonal matrix, and the columns of vectors is the eigenvectors of. The Schur decomposition of is referred to as spectral decomposition of.
  • If is positive definite, then the Schur decomposition is the same as the singular value decomposition of.

Construction of a Schur decomposition

Be. First, an eigenvalue and a corresponding eigenvector must be found for. Now vectors are chosen so that form an orthonormal basis in. These vectors form the columns of a matrix with

Which is a matrix. Now, this process is repeated for. The result is a unitary matrix with

Which is a matrix. Then we have

,

Which applies with. The entire procedure is repeated n times until the matrices are available. Then is a unitary matrix and an upper triangular matrix. Thus, the Schur decomposition of the matrix is ​​determined.

Example

For example, consider the matrix with the eigenvalues ​​(the matrix is not diagonalizable because the dimension of the associated with this eigenvalue eigenspace is 1 ).

We choose as the basis for the beginning of the standard basis, where the - th unit vector called.

For we determine an eigenvector to 2, for example, with representation and extend it to a linearly independent basis, eg. From this new base, we create the basis transformation and calculate from it can be seen that.

For we determine an eigenvector to 2, for example, with representation and extend it to a linearly independent basis, eg. From this new base, we create the basis transformation and calculate.

As shown above, the base can be chosen arbitrarily, but the thing is very simple and interesting, when choosing the default base is pulled (if possible). Thus, the previous steps to change as follows:

For we determine an eigenvector to 2, for example, with representation and extend it to a linearly independent basis, eg. From this new base, we create the basis transformation and calculate from it can be seen that.

For we determine an eigenvector to 2, for example, with representation and extend it to a linearly independent basis, eg. From this new base, we create the basis transformation and calculate.

Here, the calculation of the representation of the vectors in the correct base is, so to speak intuitive and therefore less prone to error, also the final basis transformation is also a triangular matrix.

With the Gram- Schmidt orthogonalization the base transform matrix obtained can be made into a unitary matrix, as required.

Issai Schur Normal matrix Main diagonal Wolfram Research
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