# Weyr canonical form The image shows an example of a general Weyr matrix consisting of two blocks each of which is a basic Weyr matrix. The basic Weyr matrix in the top-left corner has the structure (4,2,1) and the other one has the structure (2,2,1,1).

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

$n_{1}+n_{2}+\cdots +n_{r}=n$ of $n$ with $n_{1}\geq n_{2}\geq \cdots \geq n_{r}\geq 1$ ### Example

The following is an example of a basic Weyr matrix.

and

### 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.

## The Weyr form is canonical

That the weyr form is a canonical form of a matrix is a consequence of the following result: To within permutation of basic Weyr blocks, each square matrix $A$ over an algebraically closed field is similar to a unique Weyr matrix $W$ . The matrix $W$ is called the Weyr (canonical ) form of $A$ .

## Computation of the Weyr canonical form

### Reduction of a nilpotent matrix to the Weyr form

Step 1

Step 2

Step 3

Step 4

Continue the processes of Steps 1 and 2 to obtain increasingly smaller square matrices $A_{1},A_{2},A_{3},\ldots$ and associated nvertible matrices $P_{1},P_{2},P_{3},\ldots$ until the first zero matrix $A_{r}$ is obtained.

Step 5

Step 6

$X={\begin{bmatrix}0&X_{12}&X_{13}&\cdots &X_{1,r-1}&X_{1r}\\&0&X_{23}&\cdots &X_{2,r-1}&X_{2r}\\&&&\ddots &\\&&&\cdots &0&X_{r-1,r}\\&&&&&0\end{bmatrix}}$ .

Step 7

Step 8

Step 9

Step 10

Step 11

Step 12

## Applications of the Weyr form

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

1. The Weyr form can be used to simplify the proof of Gerstenhaber’s Theorem which asserts that the subalgebra generated by two commuting $n\times n$ matrices has dimension at most $n$ .
2. A set of finite matrices is said to be approximately simultaneously diagonalizable if they can be perturbed to simultaneously diagonalizable matrices. The Weyr form is used to prove approximate simultaneous diagonalizability of various classes of matrices. The approximate simultaneous diagonalizability property has applications in the study of phylogenetic invariants in biomathematics.
3. The Weyr form can be used to simplify the proofs of the irreducibility of the variety of all k-tuples of commuting complex matrices.