Lecture 11 Linear Alg

When the geometric multiplicity is ____ than the algebraic multiplicity of this eigenvalue. The next best thing to an eigenvector is often referred to as a “____ eigenvector”.

less

generalized

Generalized vector v2: Solve

Generalized vector v3 (if algu was 3 but gemu only 1): Solve

Augmented matrix [λI - A | -v1]

Augmented matrix [λI - A | -v2]

Jordan canonical form of a matrix

• 1’s above the diagonal appear when there are not enough ____

eigenvectors

Any matrix A with all real eigenvalues is similar to a matrix in ____ form.

If A and B are similar matrices, they necessarily have the same Jordan canonical forms.

Jordan canonical

B^t =

Stability and powers of a matrix (discrete linear dynamical systems)

x(t+1) = A x(t), x(t) = A^Tx0

• If all eigenvalues of A satisfy |λi| < 1, then all discrete trajectories will tend toward __. Such a system is called a ____ system.

• If even one eigenvalue has |λi| > 1, then for almost all initial states x(0), the trajectories will ____ without bound. Such a system is called an ____ system.

• If there are complex conjugate pairs of eigenvalues, then trajectories in the invariant 2-dimensional subspace associated with that pair will exhibit ____.

  • If the modulus of the eigenvalues is greater than 1, the trajectory spirals ____.

  • If the modulus is less than 1, the trajectory spirals ____.

  • If the modulus equals 1, the trajectory moves around an ____ in that invariant plane.

0, stable

grow, unstable

rotation

outward

inward

ellipse

A real n × n matrix A is called orthogonally diagonalizable if there exists an _______ for R^n consisting of _____ of the matrix A.

orthonormal basis, eigenvectors

Spectral Theorem: A real n × n matrix A is orthogonally diagonalizable if and only if A is ____.

• the matrix of any ____ or ____ must necessarily be symmetric

symmetric (A = A^T)

orthogonal projection, reflection

How to find an an orthonormal basis of eigenvectors.

1. find eigenvalues

2. find eigenvectors

   1. If eigenvalue has gemu = 1, ____ its eigenvector into a unit vector for v1

   2. If eigenvalue has gemu = 2, choose the first normalized vector as v2. Take the _____ to find v3 then normalize it too

3. S = ___ S^-1 = ____

normalize

general eigenvector (with variables) • v2 = 0

[v1…vn], S^T

Dynamical systems with complex eigenvalues

x(t+1) = A x(t), λ_1,2= p ± iq = r (cosθ ± isinθ)

Let v+iw be eigenvector for λ p + iq

Then x(t)=

Dynamical systems with complex eigenvalues x(t) formula procedure

1. find eigenvalues (in complex form a±bi)

2. let r=√(a²+b²) and a±bi → ____

3. Find θ then let R_tθ= ____

4. For eigenvalue a+bi find kernel basis

   1. Let w = [x y] and solve ___

   2. choose convenient values of x & y to get eigenvector in form ___

5. Let S = ___

6. Solve x(t) = ___

r (cosθ ± isinθ)

[cos(tθ) sin(tθ) -sin(tθ) cos(tθ)]

[detλI-A] [w] = 0

u+iv

[ v u ]

r^t S R_tθ S^-1 x0