Notes
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Math
Discovery
Andrius Kulikauskas
 ms@ms.lt
 +370 607 27 665
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Matricos
 I thought this was the most basic object in mathematics. Note that the index set may be arbitrary, not necessarily numbers.
 Representations  A very important idea, which is that we access a deep structure (such as a division of everything) not directly, but by way of some representation. This term is used in algebra, for example, to distinguish a system (like a group) from the matrices which serve as its multiplication table.
 Polar decomposition. Square complex matrix A can always be written as A = UP where U is a unitary matrix and P is a positivesemidefinite Hermitian matrix. The eigenvalues of U all lie on the unit circle. The real analogue of U is the orthogonal matrix, whose determinant is either +1 (rotations) or 1 (reflections). U = e^iH where H is some Hermitian matrix. P has all nonnegative eigenvalues. It is the stretching of the eigenvectors. Thus every matrix A = B*e^iC where B and C have all nonnegative real eigenvalues.
 Symmetric and skewsymmetric. Every matrix A can be broken down as the sum of a skewsymmetric matrix 1/2*(AAT) and a symmetric matrix 1/2*(A+AT).
 Note that a category may be thought of as a deductive system, a directive graph, and hence a matrix. My thesis was on the combinatorics of the general matrix, which apparently is all generated by the symmetric functions of the eigenvalues of a matrix. So I am interested to see if that might be of value here. I want to show how a category arises from first principles. I imagine that perspectives may be thought of as morphisms (or functors).
 LinearAlgebra  Is the study of the basic properties of matrices and their effects.
 Mano tezė. Jeigu matricą išrašome Jordan canonical form, tai didžiausi ciklai tėra dvejetukai.
 Symplectic form is related to complexification and also the linking of losition and momentum.

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