Linear Algebra - test 4

Eigenvector

A nonzero vector X of nxn matrix A such that Ax=lambda x, a vector transformed by matrix A into a scalar multiple of itself

Eigenvalue

The scalar multiple for Ax=lambda x causing the vector to stretch or shrink but stay in the same direction.

Eigenspace

Lambda is an eigenvalue iff the equation (A-lambda I)x=0 has a non trivial solution

Is x an eigenvector for matrix A?

1. multiply Ax (should get a vector)

   1. New vector has to be a multiple of x for it to be en eigenvector

Is lambda an eigenvalue for A?

1. Subtract lambda from diagonal

2. Row reduce

   1. CANNOT be the identity matrix (no trivial solution)

Find the span of eigenspace or span of

1. Subtract lambda from diagonal

2. Row reduce

   1. Get general solution and write FV in brackets: {[], []}

The set of eigenvectors are…

Linearly independent

Zero is an eigenvalue iff A is…

not invertible (singular)

Similarity

A~B (nxn matrices) if there is an invertible matrix P such that B=P-1AP meaning they have the same characteristic polynomial and eigenvalues

Diagonizable

A square matrix is diagonalizable if A is similar to a diagonal matrix D. A=PDP-1

An nxn matrix is diagonalizable iff…

P has n LI eigenvectors as columns and D has eigenvalues for corresponding entries

An nxn matrix with n distinct eigenvalues…

is diagonalizable

How to diagonalize a matrix:

1. Find the eigenvalues

2. Find the eigenvectors

3. Build D —> put eigenvalues as main entry in diagonal matrix

4. Build P —> put eigenvectors into a matrix matching the order of D

Normalize a vector

1. calculate magnitude of vector

   1. divide original magnitude by vector

two vectors are orthogonal is u . v =

0

Orthogonal complement

The set of all vectors, z, that are orthogonal to w is the orthogonal complement of w

Orthogonal set

A set of vectors are orthogonal if each pair in the set are orthogonal

The orthogonal complement of the row space of A is the…

null space of A

The orthogonal complement of the column space of A… is the null space of A transpose.

is the null space of A

Orthonormal basis

If w (subspace) is spanned by an orthonormal set than the set is an orthonormal basis for w

An nxn matrix u has orthonormal columns iff…

UtransposeU=I

Orthogonal matrix

A square matrix G such that G-1=Gtranspose

Orthogonal basis

A basis for a subspace W of Rn that is an orthogonal set

Prove the determinant of an orthogonal matrix equals +- 1

If G is an orthogonal matrix then…

G-1=Gtranspose

GG-1=GGtranpose

det(I)=det(GGtranpose)

1=det(G)det(Gtranpose)

1=det(G)det(G)

1=det(G)squared

det(g)= +- 1