In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, in
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices.
...
, a Weyr canonical form (or, Weyr form or Weyr matrix) is a
square matrix
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied.
Square matrices are often ...
satisfying certain conditions. A square matrix is said to be ''in'' the Weyr
canonical form
In mathematics and computer science, a canonical, normal, or standard form of a mathematical object is a standard way of presenting that object as a mathematical expression. Often, it is one which provides the simplest representation of an obje ...
if the matrix satisfies the conditions defining the Weyr canonical form. The Weyr form was discovered by the
Czech
Czech may refer to:
* Anything from or related to the Czech Republic, a country in Europe
** Czech language
** Czechs, the people of the area
** Czech culture
** Czech cuisine
* One of three mythical brothers, Lech, Czech, and Rus'
Places
*Czech, ...
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
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
In linear algebra, a Jordan normal form, also known as a Jordan canonical form (JCF),
is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to so ...
.
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
''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America.
The ''American Mathematical Monthly'' is an e ...
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
Mathematical and theoretical biology, or biomathematics, is a branch of biology which employs theoretical analysis, mathematical models and abstractions of the living organisms to investigate the principles that govern the structure, development a ...
.
Definitions
Basic Weyr matrix
Definition
A basic Weyr matrix with
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
is an
matrix
of the following form: There is a
partition
Partition may refer to:
Computing Hardware
* Disk partitioning, the division of a hard disk drive
* Memory partition, a subdivision of a computer's memory, usually for use by a single job
Software
* Partition (database), the division of a ...
:
of
with
such that, when
is viewed as an
block matrix
In mathematics, a block matrix or a partitioned matrix is a matrix that is '' interpreted'' as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original mat ...
, where the
block
is an
matrix, the following three features are present:
# The main
diagonal
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ ...
blocks
are the
scalar matrices
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal ma ...
for
.
# The first
superdiagonal
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δΠ...
blocks
are full
column rank
In linear algebra, the rank of a matrix is the dimension of the vector space generated (or spanned) by its columns. p. 48, § 1.16 This corresponds to the maximal number of linearly independent columns of . This, in turn, is identical to the dime ...
matrices in
reduced row-echelon form (that is, an
identity matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere.
Terminology and notation
The identity matrix is often denoted by I_n, or simply by I if the size is immaterial o ...
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.
Relation between Weyr and Jordan forms
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.
[Sergeichuk]
"Canonical matrices for linear matrix problems"
Arxiv:0709.2485 ath.RT 2007
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
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
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, because ...
and let the distinct eigenvalues of
be
. The
Jordan–Chevalley decomposition In mathematics, the Jordan–Chevalley decomposition, named after Camille Jordan and Claude Chevalley, expresses a linear operator as the sum of its commuting semisimple part and its nilpotent part. The multiplicative decomposition expresses an inv ...
theorem states that
is
similar to a block diagonal matrix of the form
where
is a
diagonal matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal ma ...
,
is a
nilpotent matrix In linear algebra, a nilpotent matrix is a square matrix ''N'' such that
:N^k = 0\,
for some positive integer k. The smallest such k is called the index of N, sometimes the degree of N.
More generally, a nilpotent transformation is a linear transf ...
, 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 is leads to the generalized
eigenspace
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
decomposition theorem.
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
Basis may refer to:
Finance and accounting
*Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
*Basis trading, a trading strategy consisting of ...
for the
null space
In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the domain of the map which is mapped to the zero vector. That is, given a linear map between two vector spaces and , the kernel ...
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
invertible matrices
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that
:\mathbf = \mathbf = \mathbf_n \
where denotes the -by- identity matrix and the multiplicati ...
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 In mathematics, an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation. The elementary matrices generate the general linear group GL''n''(F) when F is a field. Left multiplication (pre-multip ...
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
Conjugation or conjugate may refer to:
Linguistics
* Grammatical conjugation, the modification of a verb from its basic form
* Emotive conjugation or Russell's conjugation, the use of loaded language
Mathematics
* Complex conjugation, the chang ...
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:
# 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 Phylogenetics, Phylogenetic invariants are polynomial relationships between the frequencies of various site patterns in an idealized DNA multiple sequence alignment. They have received substantial study in the field of Mathematical and theoretical b ...
in
biomathematics
Mathematical and theoretical biology, or biomathematics, is a branch of biology which employs theoretical analysis, mathematical models and abstractions of the living organisms to investigate the principles that govern the structure, development a ...
.
# 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
{{reflist
Linear algebra
Matrix theory
Matrix normal forms
Matrix decompositions