MATH 1850U Vocabulary Chapters 5 and 6

Eigenvalue

A scalar that represents how a linear transformation stretches or squishes a given vector.

Eigenvector

A non - zero vector that does not change its direction but is scaled by its corresponding scalar value.

Eigenspace

The solution space of (A-λI)x=0

Diagonalization

The process of transforming a square matrix into a diagonal matrix.

Geometric Multiplicity

If λo is an eigenvalue of nxn matrix A, then dimension of eigenspace corresponding to λo is that of λo

Algebraic Multiplicity

The number of times λ -λo appears as a factor in the characteristic polynomial of A.

Inner product

Function that associates a real number <u,v> with each pair of vectors u and v.

Real Inner Vector Space

A standard vector space which contains the added operation of the inner product, which takes two vectors and outputs a singular one.

Weighted Euclidean Inner Product

Inner product that contains weights, usually denoted by w.

Markov Chain

A system that undergoes a process of change. At any point in time, it can occupy one of a finite set of states.

Transition Probability

The probability of state i after the system was just in state j.

Steady State Vector

Represents the long term proportions of a system in each of its states.

Unit Sphere

the set of points in V that satisfy ||u|| = 1, as long as V is an inner product space.

Orthogonal Complement

The set of all vectors in inner product space that are orthogonal to the subspace.

Orthogonal Set

The set of vectors where vectors are orthogonal to each other.

Orthonormal Set

The set where the norm of each is equal to 1, and the vectors are all orthogonal to each other.

Orthogonal Basis

The basis of orthogonal vectors.

Orthonormal Basis

The basis of orthonormal vectors.

Gram Schmidt Process

Used to find the orthogonal basis of a set of vectors.