We are concerned with the question when Hom-Lie structures on a Lie algebra closed respect to Jordan product. Somewhat unexpectedly, this leads us certain questions connected Yang-Baxter equation, and decomposition of into sum subalgebras given properties.

Let B be a nilpotent matrix and suppose that its Jordan canonical form is determined by a partition λ. Then it is known that its nilpotent commutator NB is an irreducible variety and that there is a unique partition μ such that the intersection of the orbit of nilpotent matrices corresponding to μ with NB is dense in NB. We prove that map D given by D(λ) = μ is an idempotent map. This answers a...

Let α be an automorphism of the totally disconnected group G. The compact open subgroup, V , of G is tidy for α if [α(V ) : α(V ) ∩ V ] is minimised at V , where V ′ ranges over all compact open subgroups of G. Identifying a subgroup tidy for α is analogous to identifying a basis which puts a linear transformation into Jordan canonical form. This analogy is developed here by showing that commut...

Computing the ne canonical-structure elements of matrices and matrix pencils are ill-posed problems. Therefore, besides knowing the canonical structure of a matrix or a matrix pencil, it is equally important to know what are the nearby canonical structures that explain the behavior under small perturbations. Qualitative strata information is provided by our StratiGraphtool. Here, we present low...

Lemma 8.4 If C is a n n × matrix with 0 det ≠ C , then, there exists a n n × (complex) matrix B such that C e = . Proof: For any matrix C , there exists an invertible matrix P , s.t. 1 P CP J − = , where J is a Jordan matrix. If C e = , then, 1 1 1 P B P B e P e P P CP J − − − = = = . Therefore, it is suffice to prove the result when C is in a canonical form. Suppose that 1 ( , , ) s C diag C C...

1. Matrices, Vectors and their Basic Operations 1.1. Matrices 1.2. Vectors 1.3. Addition and Scalar Multiplication of Matrices 1.4. Multiplication of Matrices 2. Determinants 2.1. Square Matrices 2.2. Determinants 2.3. Cofactors and the Inverse Matrix 3. Systems of Linear Equations 3.1. Linear Equations 3.2. Cramer’s Rule 3.3. Eigenvalues of a Complex Square Matrix 3.4. Jordan Canonical Form 4....

If A is a square matrix with spectral radius less than 1 then A k 0 as k c, but the powers computed in finite precision arithmetic may or may not converge. We derive a sufficient condition for fl(Ak) 0 as k x) and a bound on [[fl(Ak)[[, both expressed in terms of the Jordan canonical form of A. Examples show that the results can be sharp. We show that the sufficient condition can be rephrased i...