mathwanderer.wordpress.com
Similarity and normal forms | My thoughts/summaries
https://mathwanderer.wordpress.com/2009/08/02/similarity-and-normal-forms
Just another WordPress.com weblog. Similarity and normal forms. August 2, 2009. This post will prove the uniqueness of Jordan form, Smith normal form and Rational canonical form in an equivalence class of similar matrices. A criterion to check similarlity of matrices using these forms is given, and a nice lemma of similarity being irrelevant to field extension is shown using rational canonical form. Matrix (over an algebraically closed field. We have seen in a previous post. Once we have put the matrix.
mathwanderer.wordpress.com
Polar decomposition | My thoughts/summaries
https://mathwanderer.wordpress.com/2009/07/30/polar-decomposition
Just another WordPress.com weblog. July 30, 2009. This post proves the existence of polar decomposition. The polar decomposition is a generalization of the polar form of complex numbers. Theorem 1 (Existence of polar decomposition). Complex matrix. Then. Is a positive semi-definite Hermitian matrix, and. Is unitary. Furthermore,. Is uniquely determined by. There exists a unique positive semi-definite square root for a positive semi-definite Hermitian matrix. For the left multiplication map by. From &rarr...
mathwanderer.wordpress.com
My thoughts/summaries | Just another WordPress.com weblog | Page 2
https://mathwanderer.wordpress.com/page/2
Just another WordPress.com weblog. July 26, 2009. This post is incomplete. It is intended to be a summary of Hartshorne 2.1 plus a bit more. From → Algebraic Geometry. Diagonalization and Jordan Form I. July 25, 2009. This post talks about the existence of Jordan form by decomposing a vector space. Into direct sum of generalized eigenspaces. From → Canonical Form. July 24, 2009. This post proves the Cayley-Hamilton for finite. Modules, and generalize it to Nakayama’s lemma. From → Linear algebra.
mathwanderer.wordpress.com
July | 2009 | My thoughts/summaries
https://mathwanderer.wordpress.com/2009/07
Just another WordPress.com weblog. Archive for July, 2009. July 30, 2009. July 28, 2009. July 28, 2009. July 26, 2009. Jordan form II: Computations. July 26, 2009. July 25, 2009. Diagonalization and Jordan Form I. July 24, 2009. Laquo; Older Entries. Soarerz on Smith Normal Form. Messi on Smith Normal Form. On Jordan form II: Computati…. Soarerz on Smith Normal Form. Ptp on Smith Normal Form. Create a free website or blog at WordPress.com. Create a free website or blog at WordPress.com.
mathwanderer.wordpress.com
Jordan form II: Computations | My thoughts/summaries
https://mathwanderer.wordpress.com/2009/07/26/jordan-form-ii-computations
Just another WordPress.com weblog. Jordan form II: Computations. July 26, 2009. This post talks about computation of Jordan form and Jordan basis, along with a few examples. In this post we focus on computing the Jordan form of a given matrix. Proposition 1 (Calculating the number of Jordan blocks/sizes). The number of corresponding Jordan blocks is. More generally, let. The number of Jordan blocks of size. This further shows that the number of Jordan blocks of size. 1 Fix an eigenvalue. 2 Find a basis of.
mathwanderer.wordpress.com
Varieties II: Quasiprojective Varieties | My thoughts/summaries
https://mathwanderer.wordpress.com/2009/08/22/varieties-ii-quasiprojective-varieties
Just another WordPress.com weblog. Varieties II: Quasiprojective Varieties. August 22, 2009. This post contains the definition of (quasi-) projective varieties, regular functions, morphisms, projective coordinate ring, the Nullstellensatz. It also contains the global section of sheaf of regular function over distinguished open sets and the fact that affine neighborhoods form a basis for the Zariski topology. Be an algebraically closed field. We want to parametrize the 1-dimensional subspaces of. Remember...
mathwanderer.wordpress.com
Varieties I: affine varieties | My thoughts/summaries
https://mathwanderer.wordpress.com/2009/08/17/varieties-i-affine-varieties
Just another WordPress.com weblog. Varieties I: affine varieties. August 17, 2009. This post contains the defintiion of affine varieties, regular functions, coordinate ring, the Nullstellensatz, and the quotient of an affine variety upon action of a finite group. Be a field. It is clear what points in. Is a subset of. That is the common zeros of a set of polynomials. The set of polynomials can be taken to be finite. This is because. As a variety,. Is more often denoted by. 1 the Zariski topology. Is an e...
mathwanderer.wordpress.com
Rational Canonical Form | My thoughts/summaries
https://mathwanderer.wordpress.com/2009/07/28/rational-canonical-form
Just another WordPress.com weblog. July 28, 2009. This post proves the existence of rational canonical form, using the structure theoerm of finite modules over PID. Another common canonical form is the rational canonical form. While Jordan form is more like a substitute for non-diagonalizable matrix and thus does not exist when the eigenvalues do not lie in the ground field, the rational canonical form exploits the properties of the characteristic/minimal polynomial. Matrix with coefficients in a field.
mathwanderer.wordpress.com
About | My thoughts/summaries
https://mathwanderer.wordpress.com/about
Just another WordPress.com weblog. I have been interested in math for a long time but I almost gave it up in the last two years. So this is a place for me to get myself back in and revise things I have learnt before. That said, the blogposts are intended to make my mind clear instead of being expository. Topics I would try to blog about/learn about recently:. Linear algebra: Kronecker product, Cholesky, SVD. AG: essentially Hartshorne chapter 1 and 2. I think I still have many things left ununderstood.