Compact Linear Transformations

Compact Notation for Finite Transformations

We describe a new notation for finite transformations. This compact notation extends the orbit-cycle notation for permutations and builds upon existing notations. It gives insight into the structure of transformations and reduces the length of expressions without increasing the number of types of symbols.

Linear Transformations

The n×m matrix AT obtained by exchanging rows and columns of A is called the transpose of A. A matrix A is said to be symmetric if A = AT . The sum of two matrices of equal size is the matrix of the entry-by-entry sums, and the scalar product of a real number a and an m× n matrix A is the m× n matrix of all the entries of A, each multiplied by a. The difference of two matrices of equal size A a...

On One-Sided Ideals of Rings of Linear Transformations

Learning Linear Transformations

We present a polynomial time algorithm to learn (in Valiant's PAC model) cubes in n? space (with general sides-not necessarily axes parallel) given uniformly distributed samples from the cube. In fact, we solve a more general problem of learning in polynomial time linear transformations in n?space. I.e., suppose x is an n?vector whose coordinates are mutually independent random variables with u...

Self Adjoint Linear Transformations

1 Definition of the Adjoint Let V be a complex vector space with an inner product < , and norm , and suppose that L : V → V is linear. If there is a function L * : V → V for which Lx, y = x, L * y (1.1) holds for every pair of vectors x, y in V , then L * is said to be the adjoint of L. Some of the properties of L * are listed below. Proof. Introduce an orthonomal basis B for V. Then find the m...