Weyr canonical form

In mathematics, in linear algebra, a Weyr canonical form (or, Weyr form or Weyr matrix) is a square matrix satisfying certain conditions. A square matrix is said to be "in" the Weyr canonical form if the matrix satisfies the conditions defining the Weyr canonical form. The Weyr form was discovered by the Czech mathematician Eduard Weyr in 1885. 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. The Weyr form has been rediscovered several times since Weyr’s original discovery in 1885. This form has been variously called as "modified Jordan form," "reordered Jordan form," "second Jordan form," and "H-form". The current terminology is credited to Shapiro who introduced it in a paper published in the American Mathematical Monthly in 1999.
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.
Definitions.
Definition.
A basic Weyr matrix with eigenvalue formula_1 is an formula_2 matrix formula_3 of the following form: There is a partition 
such that, when formula_3 is viewed as an formula_8 block matrix formula_9, where the formula_10 block formula_11 is an formula_12 matrix, the following three features are present:
In this case, we say that formula_3 has Weyr structure formula_23.
Example.
The following is an example of a basic Weyr matrix.
formula_24
formula_25
In this matrix, formula_26 and formula_27. So formula_28 has the Weyr structure formula_29. Also,
formula_30
and
formula_31
Definition.
Let formula_32 be a square matrix and let formula_33 be the distinct eigenvalues of formula_34. We say that formula_32 is in Weyr form (or is a Weyr matrix) if formula_32 has the following form:
formula_37
where formula_38 is a basic Weyr matrix with eigenvalue formula_39 for formula_40.
Example.
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.
Relation between Weyr and Jordan forms.
The Weyr canonical form formula_41 is related to the Jordan form formula_42 by a simple permutation formula_43 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.
The Weyr form is canonical.
That the Weyr form is a canonical form of a matrix is a consequence of the following result: "Each square matrix formula_44 over an algebraically closed field is similar to a Weyr matrix formula_3 which is unique up to permutation of its basic blocks. The matrix formula_3 is called the Weyr (canonical) form of formula_44."
Computation of the Weyr canonical form.
Reduction to the nilpotent case.
Let formula_44 be a square matrix of order formula_5 over an algebraically closed field and let the distinct eigenvalues of formula_44 be formula_51. The Jordan–Chevalley decomposition theorem states that formula_44 is similar to a block diagonal matrix of the form
formula_53
where formula_54 is a diagonal matrix, formula_55 is a nilpotent matrix, and formula_56, justifying the reduction of formula_55 into subblocks formula_58. So the problem of reducing formula_44 to the Weyr form reduces to the problem of reducing the nilpotent matrices formula_58 to the Weyr form. This is leads to the generalized eigenspace decomposition theorem.
Reduction of a nilpotent matrix to the Weyr form.
Given a nilpotent square matrix formula_44 of order formula_62 over an algebraically closed field formula_63, the following algorithm produces an invertible matrix formula_64 and a Weyr matrix formula_28 such that formula_66.
Step 1
Let formula_67
Step 2
Step 3
If formula_74 is nonzero, repeat Step 2 on formula_74.
Step 4
Continue the processes of Steps 1 and 2 to obtain increasingly smaller square matrices formula_89 and associated invertible matrices formula_90 until the first zero matrix formula_91 is obtained.
Step 5
The Weyr structure of formula_44 is formula_93 where formula_94 = nullityformula_95.
Step 6
Step 7
Use elementary row operations to find an invertible matrix formula_101 of appropriate size such that the product formula_102 is a matrix of the form formula_103.
Step 8
Set formula_104 diag formula_105 and compute formula_106. In this matrix, the formula_107-block is formula_108.
Step 9
Find a matrix formula_109 formed as a product of elementary matrices such that formula_110 is a matrix in which all the blocks above the block formula_108 contain only formula_112's.
Step 10
Repeat Steps 8 and 9 on column formula_113 converting formula_114-block to formula_115 via conjugation by some invertible matrix formula_116. Use this block to clear out the blocks above, via conjugation by a product formula_117 of elementary matrices.
Step 11
Repeat these processes on formula_118 columns, using conjugations by formula_119. The resulting matrix formula_3 is now in Weyr form.
Step 12
Let formula_121. Then formula_122.
Applications of the Weyr form.
Some well-known applications of the Weyr form are listed below: