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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.[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.

Contents

DefinitionsEdit

Basic Weyr matrixEdit

DefinitionEdit

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:

  1. The main diagonal blocks   are the   scalar matrices   for  .
  2. The first superdiagonal blocks   are full column rank   matrices in reduced row-echelon form (that is, an identity matrix followed by zero rows) for  .
  3. All other blocks of W are zero (that is,   when  ).

In this case, we say that   has Weyr structure  .

ExampleEdit

The following is an example of a basic Weyr matrix.

     

In this matrix,   and  . So   has the Weyr structure  . Also,

 

and

 

General Weyr matrixEdit

DefinitionEdit

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  .

ExampleEdit

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 formsEdit

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.[6]

The Weyr form is canonicalEdit

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 formEdit

Reduction to the nilpotent caseEdit

Let   be a square matrix of order   over an algebraically closed field and let the distinct eigenvalues of   be  . The Jordan–Chevalley decomposition theorem states that   is similar to a block diagonal matrix of the form

 

where   is a diagonal matrix,   is a nilpotent matrix, 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 decomposition theorem.

Reduction of a nilpotent matrix to the Weyr formEdit

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

  1. Compute a basis for the null space of  .
  2. Extend the basis for the null space of   to a basis for the  -dimensional vector space  .
  3. Form the matrix   consisting of these basis vectors.
  4. Compute  .   is a square matrix of size   − nullity  .

Step 3

If   is nonzero, repeat Step 2 on  .

  1. Compute a basis for the null space of  .
  2. Extend the basis for the null space of   to a basis for the vector space having dimension   − nullity  .
  3. Form the matrix   consisting of these basis vectors.
  4. 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   until the first zero matrix   is obtained.

Step 5

The Weyr structure of   is   where   = nullity .

Step 6

  1. Compute the matrix   (here the  's are appropriately sized identity matrices).
  2. 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 formEdit

Some well-known applications of the Weyr form are listed below:[3]

  1. 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  .
  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.

ReferencesEdit

  1. ^ Eduard Weyr (1885). "Répartition des matrices en espèces et formation de toutes les espèces" (PDF). Comptes Rendus, Paris. 100: 966–969. Retrieved 10 December 2013.
  2. ^ Eduard Weyr (1890). "Zur Theorie der bilinearen Formen". Monatshefte für Mathematik und Physik. 1: 163–236.
  3. ^ a b c d Kevin C. Meara; John Clark; Charles I. Vinsonhaler (2011). Advanced Topics in Linear Algebra: Weaving Matrix Problems through the Weyr Form. Oxford University Press.
  4. ^ a b c 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.
  5. ^ Shapiro, H. (1999). "The Weyr characteristic". The American Mathematical Monthly. 106: 919–929. doi:10.2307/2589746.
  6. ^ Sergeichuk, "Canonical matrices for linear matrix problems", Arxiv:0709.2485 [math.RT], 2007