For any given matrix, the Schur decomposition (or factorization) allows us to find another matrix that is similar to the given one and is upper triangular. Moreover, the change-of-basis matrix used in the similarity transformation is unitary.
In other words, the Schur decomposition shows that each matrix is unitarily similar to an upper triangular matrix.
Let us revise some notions that are essential to understand the Schur decomposition.
We say that two square matrices and are similar if and only if there exists an invertible matrix such that
The matrix involved in the similarity transformation is called a change-of-basis matrix.
Two similar matrices have the same rank, trace, determinant and eigenvalues. Moreover, the algebraic and geometric multiplicities of their eigenvalues coincide.
We say that a square matrix is unitary if and only if its columns have unit norm and are orthogonal to each other, in which case has the property thatwhere denotes the conjugate transpose of .
When is unitary, the similarity transformation above becomesand we say that and are unitarily similar.
In the proof that any square matrix is unitarily similar to an upper triangular matrix we are going to repeatedly use so-called Householder matrices.
Householder matrices are among the most powerful tools in modern linear algebra. We therefore recommend to study them in detail. However, here is a summary of what we need to know in order to understand the proof of the Schur decomposition:
if is a column vector and is the identity matrix, then the matrixis called a Householder matrix ( is the norm of and its conjugate transpose);
a Householder matrix is unitary and ;
for any vector , we have that if and only if ;
for any non-zero vector , there is a Householder matrix such thatwhere is a non-zero scalar and is the first vector of the canonical basis (i.e., a vector that has the first entry equal to 1 and all the other entries equal to 0).
We are now ready to present the Schur decomposition.
Proposition Let be a square matrix. Then, is unitarily similar to an upper triangular matrix , that is,where is a unitary matrix.
Let be any eigenvalue of and one of its associated eigenvectors (hence, ). We can find a Householder matrix such thatwhere . Moreover,Partition so as to form a block-matrixwhere is the first column of and contains all the other columns. Since , we haveBy using the rules for the multiplication of block matrices, we obtainwhere in step we have used the fact that is an eigenvector of . We can writewhere is a vector of zeros, is a vector of possibly non-zero entries (equal to the first row of ), and is a matrix (containing all the rows of except the first one). This was the first step of the triangularization process. Let us now start the second step. Let be any eigenvalue of and one of its associated eigenvectors. We construct a Householder matrix such thatwhere . Note that is now shorter than in the previous step: its first entry is 1 and all its other entries are 0, but the number of 0s decreases by one unit. To be rigorous, we should change notation, but we keep the same notation for clarity's sake. As before, the symmetric equalityholds and, similarly to what we have demonstrated above, we haveWe create a new block-matrixThe matrix , being a Householder matrix, is unitary. The first column of has unit norm and it is orthogonal to all the other columns. Thus, is unitary. We can now compute the productwhere we have denoted various new vectors of possibly non-zero entries by . This ends the second step of the triangularization process. The successive steps are straightforward modifications of the second one. We keep doing Householder transformations until we obtainSince the product of unitary matrices is unitary, the matrixis unitary. Moreover,because all the matrices involved in the product are derived from Householder matrices (which are Hermitian), possibly by adding blocks of zeros and ones which are unaffected by conjugate transposition. Therefore,where is the upper triangular matrix above (with entries on the main diagonal).
The Schur decomposition is not unique. This can be seen easily from the algorithm used in the constructive proof above: at each step we choose an eigenvalue arbitrarily; as a consequence, there are different possible orderings of the eigenvalues of on the main diagonal of .
More in general, if is a Schur decomposition of , we can take any unitary matrix such thatis upper triangular, and use it to construct another Schur decomposition
A triangular matrix has the property that its diagonal entries are equal to its eigenvalues. Moreover, two similar matrices have the same eigenvalues. Therefore, the Schur decomposition allows us to read the eigenvalues of on the main diagonal of , which is upper triangular and similar to .
We should compare the results derived here with those presented in the lecture on matrix diagonalization, where we have shown that if has no defective eigenvalues, then it is similar to a diagonal matrix having the eigenvalues of on the main diagonal.
Here are the similarity and differences:
all matrices have a Schur decomposition, but only those that have no defective eigenvalues are diagonalizable;
in the Schur decomposition we can read the eigenvalues from an upper triangular matrix, while in a matrix diagonalization we read them from a (simpler) diagonal matrix;
in the Schur decomposition the change-of-basis matrix is unitary, while in a diagonalization it need only be invertible.
Each of the two decompositions has its own strengths. As a consequence, they are useful in different situations.
A simple example follows.
Example DefineThen, the Schur decomposition of iswhereandFrom we can read the eigenvalues of , which are and .
Please cite as:
Taboga, Marco (2021). "Schur decomposition", Lectures on matrix algebra. https://www.statlect.com/matrix-algebra/Schur-decomposition.
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