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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 positive-semidefinite 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 skew-symmetric. Every matrix A can be broken down as the sum of a skew-symmetric matrix 1/2*(A-AT) 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.

#### Matrix

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