Linear Algebra - Exam 3

Question 1

Eigenvector

Answer 1

For a nxn matrix A, x is a non-zero vector such that Ax = 𝜆x for some scalar 𝜆.

Question 2

Eigenvalue

Answer 2

A value 𝜆 of A gives a non-trivial solution vector x to Ax = 𝜆x

Question 3

What is the more-used formula for Ax = 𝜆x?

Answer 3

(A-𝜆I)x = 0

Question 4

Eigenspace of A corresponding to 𝜆

Answer 4

The set of all solutions to (A-𝜆I)x = 0 is Nul(A-𝜆I). It is a subspace of ℝn

Question 5

Characteristic Polynomial

Answer 5

P(𝜆) = det(A-𝜆I)

Question 6

Properties of the characteristic equation

Answer 6

1) 𝜆 is an eigenvalue iff det(A-𝜆I) = 0
2) If 𝜆 is an eigenvalue, Eig(𝜆) = Nul(A-𝜆I)
3) The roots of the characteristic equation give the eigenvalues of A
4) An nxn matrix can have maximum n eigenvalues

Question 7

Similar Matrices A and B

Answer 7

A and B are similar if there exists an invertible matrix P∈ ℝnxn such that A = PBP^(-1)

Question 8

Diagonalizable

Answer 8

A is diagonalizable if A = PDP^(-1), with diagonalizable D

Question 9

Properties of a Diagonalizable Matrix

Answer 9

• An nxn matrix A is diagonalizable if it has n linearly independent eigenvectors
• A = PDP^(-1) iff the columns of P are n linearly independent eigenvectors of A
• The diagonal entries of D are the eigenvalues corresponding to the eigenvectors in P
• A is diagnoalizable iff there are enough eigenvectors to form a basis of ℝn, called the eigenvector basis of ℝn

Question 10

For A with less than n linearly independent eigenvectors (Diagonalization)

Answer 10

• A has distinct eigenvalues 𝜆1, …, 𝜆p
• The geometric multiplicity of 𝜆k, with 1<k<p, is less than or equal to the algebraic multiplicity of 𝜆k
• Bk is a basis for the eigenspace corresponding to 𝜆k for each k, and the total collection of vectors in the sets B1,…,Bp forms an eigenvector basis for ℝn

Question 11

Inner Product

Answer 11

• uv**=**u**^(T)v**
• u1v1+…+unvn

Question 12

Length

Answer 12

• ||v|| = √(v)^2
• √(v1^2+v2^2+…+vn^2)
• ||v||^2 = v^2

Question 13

Normalization

Answer 13

• multiplication by 1/||v|| = unit vector u
• ||u|| = 1
• u is always in the same direction as v

Question 14

Distance

Answer 14

• dist(u,v) = ||u-v||

Question 15

Orthogonality

Answer 15

• u*v = 0
• 0 is orthogonal to all vectors in ℝn
• ||u+v|| = ||u||^2 + ||v||^2

Question 16

Orthogonal Complement (W perp)

Answer 16

W⊥ is the orthogonal complement of W if every vector in W⊥ is orthogonal to every vector in W.

Question 17

Properties of W perp

Answer 17

• x∈W⊥ iff x is orthogonal to every vector in a set that spans W
• W⊥ is a subspace of ℝn

Question 18

Pythagorean Property

Answer 18

uv** = ||**u**||||v**||*cos(𝛼), where 𝛼 is the angle between u and v

Question 19

Orthogonal Sets

Answer 19

• A set S={u1, … , up} is an orthogonal set if every distinct pair of vectors in the set is orthogonal. That is, ui*uj = 0 when i≠j.
• If a set is orthogonal, then it is also linearly independent
• S is a basis for the subspace spanned by S

Question 20

Orthogonal Basis

Answer 20

S={u1, … , up} is an orthogonal basis for W in ℝn if for every vector y in W, the weights in the linear combination y = c1u1+….+cpup are given by ci=(y*ui)/ui^2 for i = 1, … , p