Weyr canonical form

In mathematics, in linear algebra, a Weyr canonical form (or, Weyr form or Weyr matrix) is a square matrix which (in some sense) induces "nice" properties with matrices it commutes with.

It also has a particularly simple structure and the conditions for possessing a Weyr form are fairly weak, making it a suitable tool for studying classes of commuting matrices.

A square matrix is said to be in the Weyr canonical form if the matrix has the structure defining the Weyr canonical form.

[1][2][3] The Weyr form did not become popular among mathematicians and it was overshadowed by the closely related, but distinct, canonical form known by the name Jordan canonical form.

[4] The current terminology is credited to Shapiro who introduced it in a paper published in the American Mathematical Monthly in 1999.

[4][5] Recently several applications have been found for the Weyr matrix.

Of particular interest is an application of the Weyr matrix in the study of phylogenetic invariants in biomathematics.

A basic Weyr matrix with eigenvalue

of the following form: There is an integer partition such that, when

matrix, the following three features are present: In this case, we say that

The following is an example of a basic Weyr matrix.

is a basic Weyr matrix with eigenvalue

The following image shows an example of a general Weyr matrix consisting of three basic Weyr matrix blocks.

The basic Weyr matrix in the top-left corner has the structure (4,2,1) with eigenvalue 4, the middle block has structure (2,2,1,1) with eigenvalue -3 and the one in the lower-right corner has the structure (3, 2) with eigenvalue 0.

is related to the Jordan form

for each Weyr basic block as follows: The first index of each Weyr subblock forms the largest Jordan chain.

After crossing out these rows and columns, the first index of each new subblock forms the second largest Jordan chain, and so forth.

[6] That the Weyr form is a canonical form of a matrix is a consequence of the following result:[3] Each square matrix

over an algebraically closed field is similar to a Weyr matrix

which is unique up to permutation of its basic blocks.

is called the Weyr (canonical) form of

be a square matrix of order

over an algebraically closed field and let the distinct eigenvalues of

is similar to a block diagonal matrix of the form

This leads to the generalized eigenspace decomposition theorem.

, the following algorithm produces an invertible matrix

Step 4 Continue the processes of Steps 1 and 2 to obtain increasingly smaller square matrices

Step 6 Step 7 Use elementary row operations to find an invertible matrix

formed as a product of elementary matrices such that

Some well-known applications of the Weyr form are listed below:[3]