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 which (in some sense) induces "nice" properties with matrices it commutes with. It also has a particularly simple structure and the conditions for possessing a Weyr form are fairly weak, making it a suitable tool for studying classes of commuting matrices. A square matrix is said to be in the Weyr canonical form if the matrix has the structure 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.
^Eduard Weyr (1885). "Répartition des matrices en espèces et formation de toutes les espèces" (PDF). Comptes Rendus de l'Académie des Sciences de Paris. 100: 966–969. Retrieved 10 December 2013.
^Eduard Weyr (1890). "Zur Theorie der bilinearen Formen". Monatshefte für Mathematik und Physik. 1: 163–236.
^ abKevin C. Meara; John Clark; Charles I. Vinsonhaler (2011). Advanced Topics in Linear Algebra: Weaving Matrix Problems through the Weyr Form. Oxford University Press.
^ abcKevin 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" (PDF). The American Mathematical Monthly. 106 (10): 919–929. doi:10.2307/2589746. JSTOR 2589746. S2CID 56072601.
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