Advanced Topics in Linear Algebra: Weaving Matrix Problems - download pdf or read online

By Kevin O'Meara, John Clark, Charles Vinsonhaler

ISBN-10: 0199793735

ISBN-13: 9780199793730

The Weyr matrix canonical shape is a principally unknown cousin of the Jordan canonical shape. came upon by means of Eduard Weyr in 1885, the Weyr shape outperforms the Jordan shape in a few mathematical occasions, but it continues to be slightly of a secret, even to many that are expert in linear algebra. Written in an attractive kind, this ebook offers numerous complex issues in linear algebra associated during the Weyr shape. Kevin O'Meara, John Clark, and Charles Vinsonhaler advance the Weyr shape from scratch and contain an set of rules for computing it. a desirable duality exists among the Weyr shape and the Jordan shape. constructing an figuring out of either types will let scholars and researchers to use the mathematical features of every in various events. Weaving jointly rules and purposes from a number of mathematical disciplines, complicated issues in Linear Algebra is way greater than a derivation of the Weyr shape. It provides novel purposes of linear algebra, similar to matrix commutativity difficulties, approximate simultaneous diagonalization, and algebraic geometry, with the latter having topical connections to phylogenetic invariants in biomathematics and multivariate interpolation. one of the comparable mathematical disciplines from which the booklet attracts principles are commutative and noncommutative ring conception, module conception, box idea, topology, and algebraic geometry. a variety of examples and present open difficulties are incorporated, expanding the book's application as a graduate textual content or as a reference for mathematicians and researchers in linear algebra.

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Additional info for Advanced Topics in Linear Algebra: Weaving Matrix Problems Through the Weyr Form

Example text

Then the subspaces Wi are invariant under T, and V is their direct sum : V = W1 ⊕ W2 ⊕ · · · ⊕ Wk . Proof In essence, the proof is the same one used to establish the primary decomposition of a finite abelian group into p-groups. 1, the Wi are T-invariant. i For each i, let fi = p/pm i . Since these polynomials are relatively prime, there exist polynomials g1 , g2 , . . , gk such that f1 g1 + f2 g2 + · · · + fk gk = 1. It follows that f1 (T)g1 (T) + f2 (T)g2 (T) + · · · + fk (T)gk (T) = I . Now given v ∈ V , we have v = I(v) = ki=1 fi (T)gi (T)(v) with fi (T)gi (T)(v) ∈ Wi because pi (T)mi fi (T)gi (T)(v) = p(T)gi (T)(v) = 0.

If A is nilpotent then 0 is its only eigenvalue. 1 Over an algebraically closed field, an n × n matrix A is nilpotent if and only if 0 is the only eigenvalue of A. Also, a square matrix that does not have two distinct eigenvalues must be the sum of a scalar matrix λI and a nilpotent matrix. Proof The second statement follows from the first because if λ is the only eigenvalue of A, then 0 is the only eigenvalue of A − λI (and A = λI + (A − λI) ). Suppose 0 is the only eigenvalue of A. Since the field is algebraically closed, the characteristic polynomial of A must be p(x) = xn .

A good canonical form under similarity allows an indirect way of seeing if two given matrices A, B ∈ Mn (F) are similar, by checking if they have the same canonical form. This is sometimes highlighted by authors as the raison d’être for having a canonical form. That the present authors chose not to list this outcome among their five desirable features of a canonical form would suggest they disagree ! Nevertheless, the Jordan and Weyr forms both perform well in this method of testing for similarity of matrices A and B (with perhaps Weyr a little more transparent), providing one knows the eigenvalues of A and B (which must agree for similar matrices), and also knows the nullities of the various powers (λI − A)i and (λI − B)i for i = 1, 2, .

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Advanced Topics in Linear Algebra: Weaving Matrix Problems Through the Weyr Form by Kevin O'Meara, John Clark, Charles Vinsonhaler


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