TheWeyrCanonicalForm Robert M. Corless and Steven E. Thornton Ontario Research Centre for Computer Algebra, Western University Introduction Weyr Canonical Form Example The Jordan canonical form is a well known and highly studied matrix canonical Theorem 1 Every square matrix A over an algebraically closed field F is similar to a Consider the 8 × 8 matrix J in Jordan canonical form with Jordan structure (3; 2; 2; 1) form. This poster discusses its younger, less popular cousin, the Weyr canonical form. matrix in Weyr form (see [1] for a proof). 2 3 Originally discovered in 1885 by Czech mathematician Eduard Weyr [3], the Weyr 2 1 0 6 7 form has had difficulty gaining the attention of mathematicians. Some applications Corollary 1 For every square matrix A over an algebraically closed field F , there exists 6 2 1 7 −1 6 7 that are unique to the Weyr form include: “commutativity problems, approximate an invertible matrix Q such that W = QAQ gives a Weyr matrix, known as the Weyr 6 7 6 2 7 simultaneous diagonalization, and algebraic geometry” [1]. canonical form of A. 6 7 6 2 1 7 6 7 J = 6 7 : This poster focuses on the computation of the Weyr canonical form though first com- Corollary 2 The Weyr canonical form of a matrix is unique up to permutations of the 6 2 7 6 7 puting the Jordan canonical form. An algorithm can be found in Chapter 2.5 of [1] basic Weyr matrices. That is, there is no required ordering for the basic Weyr matrices 6 2 1 7 W in equation (1). 6 7 for the direct computation of the Weyr form. When we implemented this algorithm it i 4 2 5 was found to be too expensive to be of practical use. We discuss a simple alternative 2 below. Duality between the Jordan and Weyr Forms We will find a permutation matrix P such that the Weyr form W of J is given by Let a basic Jordan matrix be the direct sum of Jordan block matrices where each W = PJP −1. Weyr Matrix block has the same eigenvalue and the blocks are arranged in order of decreasing size along the diagonal. Basic Weyr Matrix We can view J as the matrix of a linear transformation T of C8 relative to some × (ordered) basis B = fv1; v2; v3; v4; v5; v6; v7; v8g. The action of T on the basis vectors An n n matrix W is a basic Weyr matrix with eigenvalue λ if, for integers ni, The Jordan structure of a basic Jordan matrix is the sequence of the sizes of the 1 ≤ i ≤ r, where n + n + ··· + n = n and n ≥ n ≥ · · · ≥ n ≥ 1, the following can be represented as: 1 2 r 1 2 r Jordan blocks along the diagonal. properties hold. 0 v1 v2 v3 0 v4 v5 Each partition (n1; n2; : : : ; nr) of n determines a Young tableau (or Ferrer’s diagram): • It can be viewed as a block upper bidiagonal matrix with r diagonal blocks 0 v6 v7 (r − 1 super-diagonal block). 0 v8 ··· n1 boxes × If we reorder the basis vectors in B by running through them in column order, and • Each of the diagonal blocks are of the form λI where I is an ni ni identity ··· n2 boxes taking the vectors v to be the standard basis vectors, we get the basis matrix. i . 82 3 2 3 2 3 2 3 2 3 2 3 2 3 2 39 T • The super-diagonal blocks Wi;i+1 are of the form [I 0] where I is an ni+1 × ··· > 1 0 0 0 0 0 0 0 > nr boxes >6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7> ni+1 identity matrix and 0 is a zero matrix of size (ni − ni+1) × ni+1. >6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7> >607 607 607 607 617 607 607 607> “Transposing” the tableau gives a Young tableau that corresponds to the dual partition >6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7> • All other blocks are zero. >607 607 607 607 607 607 607 617> (m ; m ; : : : ; m ) of (n ; n ; : : : ; n ). In terms of the original tableau, m is the number <>6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7=> 1 2 s 1 2 r 1 607 617 607 607 607 607 607 607 of boxes in the first column (= r), m is the number in the second column, and so on. B0 f g 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 2 = v1; v4; v6; v8; v2; v5; v7; v3 = >6 7 ; 6 7 ; 6 7 ; 6 7 ; 6 7 ; 6 7 ; 6 7 ; 6 7> : Properties of a Basic Weyr Matrix For example, >607 607 607 607 607 617 607 607> >6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7> >607 607 617 607 607 607 607 607> • The eigenvalue λ has algebraic multiplicity n and geometric multiplicity >6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7> >4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5> n . > 0 0 0 0 0 0 1 0 > 1 and : ; 0 0 0 1 0 0 0 0 • The Weyr structure (also known as the Weyr characteristic [2]) of a basic Weyr matrix is the sequence (n1; n2; : : : ; nr). Rewriting the basis vectors in a permutation matrix P gives are the tableaux corresponding to the dual partitions (5; 3; 2) and (3; 3; 2; 1; 1). 2 3 2 3 1 0 0 0 0 0 0 0 2 0 0 0 1 0 0 Examples 6 7 6 7 Theorem 2 The Weyr and Jordan structures of an n×n matrix with a single eigenvalue 6 7 6 7 60 0 0 0 1 0 0 07 6 0 2 0 0 0 1 0 7 are dual partitions of n. 6 7 6 7 Weyr structure (3; 2; 2) Weyr structure (3; 3; 1) 60 0 0 0 0 0 0 17 6 0 0 2 0 0 0 1 7 2 3 2 3 6 7 6 7 60 1 0 0 0 0 0 07 −1 6 0 0 0 2 0 0 0 7 λ 0 0 1 0 λ 0 0 1 0 0 P = 6 7 : Thus W = PJP = 6 7 : 6 7 6 7 The General Case 60 0 0 0 0 1 0 07 6 2 0 0 1 7 6 7 6 7 6 7 6 7 6 0 λ 0 0 1 7 6 0 λ 0 0 1 0 7 ∗ 6 7 6 7 6 7 6 7 Let J be a matrix in Jordan form. There exists a matrix J similar to J, also in 60 0 1 0 0 0 0 07 6 0 2 0 0 7 6 0 0 λ 0 0 7 6 0 0 λ 0 0 1 7 ∗ ∗ ∗ 6 7 6 7 6 7 6 7 Jordan form, where J is the direct sum of basic Jordan matrices Ji . Each Ji has one 40 0 0 0 0 0 1 05 4 0 0 2 0 5 6 7 6 7 6 6 ∗ 6 λ 0 1 0 7 6 λ 0 0 1 7 eigenvalue λi where λi = λj for i = j. Let Pi be the Jordan structure of J and let Qi 0 0 0 1 0 0 0 0 2 6 7 6 7 6 7 6 7 be the dual of Pi. We can define Wi to be the basic Weyr matrix with eigenvalue λi 6 0 λ 0 1 7 6 0 λ 0 0 7 6 7 6 7 and Weyr structure Qi. The matrix W = diag(W1;W2;::: ) is the Weyr form of J. 4 λ 0 5 4 0 0 λ 0 5 Implementation 0 λ λ Example An implementation in the Maple computer algebra system for computing the Weyr form Both J and J ∗ are in Jordan form with J similar to J ∗. J ∗ is the direct sum of two of a matrix can be found at: github.com/StevenThornton/WeyrForm Weyr Matrix basic Jordan matrices. Computing the Weyr form of J is equivalent to computing the ∗ A matrix W with eigenvalues λ1; : : : ; λk, λi =6 λj for i =6 j is in Weyr form (and is a Weyr form of J . This can be done by computing the basic Weyr matrices that are ∗ ∗ References Weyr matrix) if it is the direct sum of basic Weyr matrices, where Wi has eigenvalue similar to each basic Jordan matix in J . The Weyr form of J and J is W . λi. It has the form: 2 3 2 3 2 3 2 3 [1] Kevin O’Meara et al. Advanced Topics in Linear Algebra: Weaving Matrix Problems 5 5 1 5 0 0 1 W1 6 7 6 7 6 7 through the Weyr Form. Oxford University Press, USA, 2011. 6 7 6 2 1 7 6 5 7 6 0 5 0 0 7 6 7 6 7 6 7 6 7 6 W2 7 6 2 7 6 5 7 6 0 0 5 0 7 [2] Helene Shapiro. The Weyr characteristic.