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. 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
Basic Weyr matrix
Definition A basic Weyr matrix with eigenvalue λ {\displaystyle \lambda } is an n × n {\displaystyle n\times n} matrix W {\displaystyle W} of the following form: There is an integer partition
n 1 + n 2 + ⋯ + n r = n {\displaystyle n_{1}+n_{2}+\cdots +n_{r}=n} of n {\displaystyle n} with n 1 ≥ n 2 ≥ ⋯ ≥ n r ≥ 1 {\displaystyle n_{1}\geq n_{2}\geq \cdots \geq n_{r}\geq 1}
such that, when W {\displaystyle W} is viewed as an r × r {\displaystyle r\times r} block matrix ( W i j ) {\displaystyle (W_{ij})} , where the ( i , j ) {\displaystyle (i,j)} block W i j {\displaystyle W_{ij}} is an n i × n j {\displaystyle n_{i}\times n_{j}} matrix, the following three features are present:
The main diagonal blocks W i i {\displaystyle W_{ii}} are the n i × n i {\displaystyle n_{i}\times n_{i}} scalar matrices λ I {\displaystyle \lambda I} for i = 1 , … , r {\displaystyle i=1,\ldots ,r} . In other words, the entries W i i {\displaystyle W_{ii}} in the main block are eigenvalues. The first superdiagonal blocks W i , i + 1 {\displaystyle W_{i,i+1}} are full column rank n i × n i + 1 {\displaystyle n_{i}\times n_{i+1}} matrices in reduced row-echelon form (that is, an identity matrix followed by zero rows) for i = 1 , … , r − 1 {\displaystyle i=1,\ldots ,r-1} . This is equivalent to an identity matrix in reduced row echelon form, above the main blocks. All other blocks of W are zero (that is, W i j = 0 {\displaystyle W_{ij}=0} when j ≠ i , i + 1 {\displaystyle j\neq i,i+1} ). In this case, we say that W {\displaystyle W} has Weyr structure ( n 1 , n 2 , … , n r ) {\displaystyle (n_{1},n_{2},\ldots ,n_{r})} .
Example The following is an example of a basic Weyr matrix.
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