Let V be a finite-dimensional F -vector space, and let T : V → V be a linear transformation. In this note we prove that V is a direct sum of cyclic T -invariant subspaces. More specifically, we prove that V is a direct sum of cyclic T -invariant subspaces whose annihilators are generated by powers of irreducible polynomials, and that the collection of these polynomials is uniquely determined. R...

8 Matrices 16 8.1 Representation of vector spaces and linear transformations . . . . . . . . . . . . . . . 16 8.2 Similarity transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 8.3 Direct sums of matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 8.4 The companion matrix of a polynomial . . . . . . . . . . . . . . . . . . . . . . . ....

in the 1960's noboru iwahori and hideya matsumoto, eiichi abe and kazuo suzuki, and michael stein discovered that chevalley groups $g=g(phi,r)$ over a semilocal ring admit remarkable gauss decomposition $g=tuu^-u$, where $t=t(phi,r)$ is a split maximal torus, whereas $u=u(phi,r)$ and $u^-=u^-(phi,r)$ are unipotent radicals of two opposite borel subgroups $b=b(phi,r)$ and $b^-=b^-(phi,r)$ contai...

In [9] Jackowski and McClure gave a homotopy decomposition theorem for the classifying space of a compact Lie group G; their theorem states that for any prime p the space BG can be constructed at p as the homotopy direct limit of a specific diagram involving the classifying spaces of centralizers of elementary abelian p-subgroups of G. In this paper we will prove a parallel algebraic decomposit...

Abstract: It is shown that the polar decomposition theorem of operators in (real) Hilbert spaces gives rise to the known decomposition in boost and spatial rotation part of any matrix of the orthochronous proper Lorentz group SO(1, 3)↑. This result is not trivial because the polar decomposition theorem is referred to a positive defined scalar product while the Lorentz-group decomposition theore...

We present fully formalized proofs of some central theorems from combinatorics. These are Dilworth’s decomposition theorem, Mirsky’s theorem, Hall’s marriage theorem and the Erdős-Szekeres theorem. Dilworth’s decomposition theorem is the key result among these. It states that in any finite partially ordered set (poset), the size of a smallest chain cover and a largest antichain are the same. Mi...

In [BMR05] the authors proved a theorem of cell decomposition for the theory CODF of closed ordered differential fields which generalize the usual Cell Decomposition Theorem for o-minimal structures. As a consequence of this result, a particularly well-behaving dimension function on definable sets in CODF was introduced. Here we carry on the study of this cell decomposition in CODF by proving t...

We give a decomposition theorem for signed graphs whose frame matroids are binary and a decomposition theorem for signed graphs whose frame matroids are quaternary.

We introduce the concept of combed graphs and present an ear decomposition theorem for this class of graphs. This theorem includes the well known ear decomposition theorem for matching covered graphs proved by Lovász and Plummer. Then we use the ear decomposition theorem to show that any two edges of a 2-connected combed graph lie in a balanced circuit of an equivalent combed graph. This result...