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).   Originally discovered in 1885 by Czech mathematician Eduard Weyr [3], the Weyr 2 1 0   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  2 1  −1   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    2  simultaneous diagonalization, and algebraic geometry” [1]. canonical form of A.    2 1    J =   . 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  2    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  2 1  W in equation (1).   for the direct computation of the Weyr form. When we implemented this algorithm it i  2  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 = {v1, v2, v3, v4, v5, v6, v7, v8}. 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 . .                 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                 ni+1 identity matrix and 0 is a zero matrix of size (ni − ni+1) × ni+1.                 0 0 0 0 1 0 0 0 “Transposing” the tableau gives a Young tableau that corresponds to the dual partition                 • All other blocks are zero. 0 0 0 0 0 0 0 1 (m , m , . . . , m ) of (n , n , . . . , n ). In terms of the original tableau, m is the number                 1 2 s 1 2 r 1 0 1 0 0 0 0 0 0 of boxes in the first column (= r), m is the number in the second column, and so on. B′ { }                 2 = v1, v4, v6, v8, v2, v5, v7, v3 =   ,   ,   ,   ,   ,   ,   ,   . Properties of a Basic Weyr Matrix For example, 0 0 0 0 0 1 0 0                 0 0 1 0 0 0 0 0 • The eigenvalue λ has algebraic multiplicity n and geometric multiplicity                                 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).     1 0 0 0 0 0 0 0 2 0 0 0 1 0 0 Examples     Theorem 2 The Weyr and Jordan structures of an n×n matrix with a single eigenvalue     0 0 0 0 1 0 0 0  0 2 0 0 0 1 0  are dual partitions of n.     Weyr structure (3, 2, 2) Weyr structure (3, 3, 1) 0 0 0 0 0 0 0 1  0 0 2 0 0 0 1          0 1 0 0 0 0 0 0 −1  0 0 0 2 0 0 0  λ 0 0 1 0 λ 0 0 1 0 0 P =   . Thus W = PJP =   .     The General Case 0 0 0 0 0 1 0 0  2 0 0 1           0 λ 0 0 1   0 λ 0 0 1 0  ∗         Let J be a matrix in Jordan form. There exists a matrix J similar to J, also in 0 0 1 0 0 0 0 0  0 2 0 0   0 0 λ 0 0   0 0 λ 0 0 1  ∗ ∗ ∗         Jordan form, where J is the direct sum of basic Jordan matrices Ji . Each Ji has one 0 0 0 0 0 0 1 0  0 0 2 0      ̸ ̸ ∗  λ 0 1 0   λ 0 0 1  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         be the dual of Pi. We can define Wi to be the basic Weyr matrix with eigenvalue λi  0 λ 0 1   0 λ 0 0      and Weyr structure Qi. The matrix W = diag(W1,W2,... ) is the Weyr form of J.  λ 0   0 0 λ 0  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 ≠ λj for i ≠ 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:         [1] Kevin O’Meara et al. Advanced Topics in Linear Algebra: Weaving Matrix Problems 5 5 1 5 0 0 1 W1       through the Weyr Form. Oxford University Press, USA, 2011.    2 1   5   0 5 0 0           W2   2   5   0 0 5 0  [2] Helene Shapiro. The Weyr characteristic. The American mathematical monthly, W =   (1)   ∗      ..  J =  5 1  J =  5  W =  5  106(10):919–929, 1999.  .         5   2 1   2 0 1  Wk  2   2   0 2 0  [3] Eduard Weyr. Répartition des matrices en espèces et formation de toutes les espèces. 5 2 2 CR Acad. Sci. Paris, 100:966–969, 1885.