Lecture 9 linear alg

If A is an n × n matrix, we call a vector v an eigenvector of A if T(v) = Av = ___ for some scalar λ. This scalar λ is called the ____ associated with the eigenvector.

λv

eigenvalue

The set of all eigenvalues of a matrix A is called the ____ of A

If an eigenvalue λ occurs as a repeated root of the characteristic polynomial, we refer to the multiplicity of the root as the ____ of the eigenvalue.

spectrum

algebraic multiplicity

If A is an n × n matrix, then p_A(λ) = _____ is an nth degree polynomial in λ called the characteristic polynomial of A. The eigenvalues are therefore the ___ of the characteristic polynomial.

det(λI − A)

roots

If λ is an eigenvalue of A, then ____ is a subspace called the eigenspace of λ, or __. As with any subspace, it is closed under scaling and vector addition.

• Corollary 1: If v is an eigenvector associated with an eigenvalue λ, then t v is also an eigenvector for any scalar t.

• Corollary 2: If v₁ and v₂ are eigenvectors associated with the same eigenvalue λ, then c₁v₁ + c₂v₂ is also an eigenvector for any scalars c₁, c₂.

ker(λI − A), E_λ

Consider an eigenvalue λ. The geometric multiplicity of E_λ of a matrix A is ____, i.e., the number of linearly independent ____ associated with this ____.

dim(ker(λI − A))

eigenvectors, eigenvalue

how to find eigenvalues of a matrix

1. calculate ____

2. find characteristic polynomial p_A(λ): ____

3. set equal to __ solve to get eigenvalues λ

how to find eigenvectors:

4. plug each value of __ into λI − A

5. compute ___ for each λ

λI − A

det(λI − A)

0

λ

kernel basis of (λI − A)

Powers of a matrix: If a matrix A is diagonalizable, we can write [A]_B = S⁻¹ A S = D for some change of basis matrix S. Therefore A = ____ and A^t = ____, where D^t = ____

S D S⁻¹

S D^t S⁻¹

consider nxn matrix A

• Eigenvectors corresponding to distinct eigenvalues are linearly ___.

• Matrix A is diagonalizable if the geometric multiplicities of the eigenvalues add up to __ (aka if it has n distinct eigenvalues)

independdent

n

Discrete dynamical systems

1. Write system as:_____

2. find eigenvalues and eigenvectors of A

3. Write v(t) = ____ = _____

v(t+1)=Av(t)

A^tv0, c₁λ₁^tv₁ + c₂λ₂^tv₂ with eigenvalues λ and eigenvectors v

Strategy for Diagonalization

1. Find the _____ of A by solving the characteristic equation _____ = 0

2. For each eigenvalue λ, find a ___ of the eigenspace

   Eλ = ____.

3. Matrix A is diagonalizable if the dimensions of the _____ add up to n. In this case, we find an eigenbasis {v1,..., vn} for A by concatenating the ___ of the eigenspaces we found in part b.

   1. S= ____, B= _____

   2. write the diagonalization of A by A=SDS^-1

eigenvalues, fA(λ) = det(A − λIn),

basis, ker(A − λIn)

eigenspaces, bases