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. The Weyr form was discovered by the Czech mathematician Eduard Weyr in 1885. [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. [3] The Weyr form has been rediscovered several times since Weyr’s original discovery in 1885. [4] This form has been variously called as modified Jordan form,reordered Jordan form,second Jordan form, and H-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 is an matrix of the following form: There is an integer partition
such that, when is viewed as an block matrix , where the block is an matrix, the following three features are present:
In this case, we say that has Weyr structure .
The following is an example of a basic Weyr matrix.
In this matrix, and . So has the Weyr structure . Also,
and
Let be a square matrix and let be the distinct eigenvalues of . We say that is in Weyr form (or is a Weyr matrix) if has the following form:
where is a basic Weyr matrix with eigenvalue for .
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.
The Weyr canonical form is related to the Jordan form by a simple permutation 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. The matrix is called the Weyr (canonical) form of .
Let be a square matrix of order over an algebraically closed field and let the distinct eigenvalues of be . The Jordan–Chevalley decomposition theorem states that is similar to a block diagonal matrix of the form
where is a diagonal matrix, is a nilpotent matrix, and , justifying the reduction of into subblocks . So the problem of reducing to the Weyr form reduces to the problem of reducing the nilpotent matrices to the Weyr form. This leads to the generalized eigenspace decomposition theorem.
Given a nilpotent square matrix of order over an algebraically closed field , the following algorithm produces an invertible matrix and a Weyr matrix such that .
Step 1
Let
Step 2
Step 3
If is nonzero, repeat Step 2 on .
Step 4
Continue the processes of Steps 1 and 2 to obtain increasingly smaller square matrices and associated invertible matrices until the first zero matrix is obtained.
Step 5
The Weyr structure of is where = nullity.
Step 6
Step 7
Use elementary row operations to find an invertible matrix of appropriate size such that the product is a matrix of the form .
Step 8
Set diag and compute . In this matrix, the -block is .
Step 9
Find a matrix formed as a product of elementary matrices such that is a matrix in which all the blocks above the block contain only 's.
Step 10
Repeat Steps 8 and 9 on column converting -block to via conjugation by some invertible matrix . Use this block to clear out the blocks above, via conjugation by a product of elementary matrices.
Step 11
Repeat these processes on columns, using conjugations by . The resulting matrix is now in Weyr form.
Step 12
Let . Then .
Some well-known applications of the Weyr form are listed below: [3]