Glossary

Eigen Decomposition

Let P be a matrix os eigenvectors of a given square matrix A, and D be a diagonal matrix with corresponding eigenvalues on the diagonal. Then, as long as P is a square matrix, A can be written as an eigen decomposition:

				A = P*D*P^-1

This is also referred to as matrix diagonalization. Here, matrix A is said to be diagonalizable. Any nxn matrix is diagonalizable as long as P is invertible. (from www.mathworld.com)

Length (of a vector)

If a vector v = (x, y, z), then the length of v (denoted ||v||) can be found by calculating:
This is also called the norm of a vector.

Orthogonal

Two vectors are orthogonal, or perpendicular, if their inner product (or dot product) is zero.

Polysemy

The existence of two or more possible meanings for a word or phrase.
ex. Second can mean a unit of time or denote an item's place in a list.

Rank

The rank of a matrix A is the number of linearly independent rows or columns of A.

Synonymy

The existence of two or more terms for the same item or idea.
ex. Car and Automobile are two different words for the same thing.

Vector Space

A vector space V is a set that is closed under finite vector addition and scalar multiplication. In order for V to be a vector space, the following conditions must hold for elements X, Y, Z in V and scalars r and s:

1. Commutativity: X+Y=Y+X
2. Associativity of vector addition: (X+Y)+Z=X+(Y+Z)
3. Additive identity: X+0=0+X=X
4. Existence of an additive inverse: For any X, there exists a (-X) such that X+(-X)=0.
5. Associativity of scalar multiplication: r(sX)=(rs)X
6. Distributivity of scalar sums: (r+s)X=rX+sX
7. Distributivity of vector sums: r(X+Y)=rX+rY
8. Scalar multiplication identity

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