Linear Algebra Test 3 (Ch. 5-6)

Eigenvector

A nonzero vector x such that Ax=lambda x for some scalar lambda

Eigenvalue

A scalar lambda of A if there is a nonzero solution x of Ax = lambda x

Eigenspace

The subspace that is the solutions of (A-lambda I)x =0

Characteristic Equation

det(A-lambda I)=0, a scalar lambda is only an eigenvalue of an n x n matrix A if lambda satisfies the characteristic equation

Characteristic Polynomial

The expression det(A-lambda I) that is an nth order polynomial in the variable lambda

Diagonalizable Matrix

An n x n matrix A is diagonalizable if it is similar to a diagonal matrix D. In other words, there is an invertible matrix P such that P-1AP=D

Inner/Dot product of two vectors

The number u transpose v, denoted u•v

Length of a vector

The nonnegative scalar defined by sqrt(v•v)

Orthogonal complement

The set of all vectors z orthogonal to a subspace W, which is also a subspace

Orthonormal set

An orthogonal set of length 1 vectors

The normal equations for Ax=b

AtAx=Atb

Least squares error

The distance from b to Axhat