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.[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.
Definitions
Basic Weyr matrix
Definition
A basic Weyr matrix with eigenvalue is an
matrix
of the following form: There is a partition
-
of
with
such that, when is viewed as an
blocked matrix
, where the
block
is an
matrix, the following three features are present:
- The main diagonal blocks
are the
scalar matrices
for
.
- The first superdiagonal blocks
are full column rank
matrices in reduced row-echelon form (that is, an identity matrix followed by zero rows) for
.
- All other blocks of W are zero (that is,
when
).
In this case, we say that has Weyr structure
.
Example
The following is an example of a basic Weyr matrix.
In this matrix, and
. So
has the Weyr structure
. Also,
and
General Weyr matrix
Definition
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
.
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:[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
.
Computation of the Weyr canonical form
Reduction to the nilpotent case
Let be a square matrix of order
over an algebraically closed field and let the distinct eigenvalues of
be
. As a consequence of the generalized eigenspace decomposition theorem, one can show that
is similar to a block diagonal matrix of the form
where is a diagonal matrix and
is a nilpotent matrix. So the problem of reducing
to the Weyr form reduces to the problem of reducing the nilpotent matrices
to the Weyr form.
Reduction of a nilpotent matrix to the Weyr form
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
- Compute a basis for the null space of
.
- Extend the basis for the null space of
to a basis for the
-dimensional vector space
.
- Form the matrix
consisting of these basis vectors.
- Compute
.
is a square matrix of size
− nullity
.
Step 3
If is nonzero, repeat Step 2 on
.
- Compute a basis for the null space of
.
- Extend the basis for the null space of
to a basis for the vector space having dimension
− nullity
.
- Form the matrix
consisting of these basis vectors.
- Compute
.
is a square matrix of size
− nullity
− nullity
.
Step 4
Continue the processes of Steps 1 and 2 to obtain increasingly smaller square matrices and associated nvertible matrices
until the first zero matrix
is obtained.
Step 5
The Weyr structure of is
where
= nullity
.
Step 6
- Compute the matrix
(here the
's are appropriately sized identity matrices).
- Compute
.
is a matrix of the following form:
-
.
-
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
.
Applications of the Weyr form
Some well-known applications of the Weyr form are listed below:[3]
- The Weyr form can be used to simplify the proof of Gerstenhaber’s Theorem which asserts that the subalgebra generated by two commuting
matrices has dimension at most
.
- 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.
- The Weyr form can be used to simplify the proofs of the irreducibility of the variety of all k-tuples of commuting complex matrices.
References
- ↑ Eduard Weyr (1985). "Répartition des matrices en espèces et formation de toutes les espèces" (PDF). Comptes Rendus, Paris 100: 966–969. Retrieved 10 December 2013.
- ↑ Eduard Weyr (1890). "Zur Theorie der bilinearen Formen". Monatsh. Math. Physik 1: 163–236.
- 1 2 3 4 Kevin C. Meara, John Clark, Charles I. Vinsonhaler (2011). Advanced Topics in Linear Algebra: Weaving Matrix Problems through the Weyr Form. Oxford University Press.
- 1 2 3 Kevin C. Meara, John Clark, Charles I. Vinsonhaler (2011). Advanced Topics in Linear Algebra: Weaving Matrix Problems through the Weyr Form. Oxford University Press. pp. 44, 81–82.
- ↑ Shapiro, H. (1999). "The Weyr characteristic". The American Mathematical Monthly 106: 919–929. doi:10.2307/2589746.