Linear Algebra Exam Three

Coordinate Systems

Suppose the set B = {b1, b2, …, bp} is a basis for a subspace H. For each x in H, x = c1b1 + c2b2 + … + cpbp, the coordinates of x relative to the basis B are the weights c1, c2, …, cp

Coordinate Vector

The vector in R^p [x]B =

c1
c2
…

cp

is called ______ of x relative to B or B-coordinate vector of x

Coordinate Mapping (Linear Transformation)

x |→ [x]_B

Theorem 4.15

Let B = {b1, b2, …, bp} and C = {c1, c2, …, cp} be bases of a vector space V. Then there is a unique n x n matrix C ← B (with a P above the arrow), such that [x]_C = C ← B [x]_B (with a P above the arrow), the columns of C ← B (with a P above the arrow) are the C-coordinate vectors of the vectors in the basis B

C ← B (with a P above the arrow) = [[b1]_C [b2]_C … [bp]_C ]

^Change-of-coordinates matrix from B to C

Applications of Eigenvectors and Eigenvalues

Web ranking system design, image compression

Eigen-

Adopted from the German word “eigen” for “own,” “proper,” or “characteristic” (chapter 5.2, the characteristic equation)

Eigenvector

An _____ of an n x n matrix A is a nonzero vector x such that Ax = λx for some scalar λ

Eigenvalue

A scalar λ is called a ______ of A if there is a nontrivial solution x of Ax = λx, such that x is called an eigenvector corresponding to λ

All the solutions of (A - λI)x = 0 is the ______ of A - λI

Null space

Eigenspace

Nul(A - λI) is a subspace of Rⁿ and is called the ______ of a A corresponding to λ

Theorem 5.1

The eigenvalues of a triangular matrix are the entries on its main diagonal.

λ is an eigenvalue of A ←> (A - λI)x = 0 has nontrivial solution ←> A - λI is not invertible ←> det(A - λI) = 0

det(A - λI) = the product of the entries on the main diagonal since A - det(A - λI)I is also a triangular matrix

Theorem 5.2

If v1, …, vr are eigenvectors corresponding to distinct eigenvalues λ1, …, λr of an n x n matrix A, then {v1, …, vr} is linearly independent

← is false

5.2 The Characteristic Equation

Ax = λx ←> (A - λI)x = 0
Has nontrivial solution → (A - λI) is not invertible

→ det(A - λI) = 0

Characteristic Equation

det(A - λI) = 0

Characteristic Polynomial

det(A - λI)

A scalar λ is an _____ of an n x n matrix A if and only if, det(A - λI) = 0

eigenvalue

Algebraic Multiplicity

This of an eigenvalue λ is its multiplicity as a root of the characteristic equation

Similarity

If A and B are n x n matrix and there exists an invertible matrix P such that P^-1 A P = B → A is similar to B

A = PBP^-1 , let Q = P^-1

→ A = Q^-1 B Q

→ B is similar to A
A |→ P^-1 A P is called _______ transformation

Theorem 5.4

If n x n matrix A and B are similar, then they have the same characteristic polynomial and hence same eigenvalues (with the same multiplicity)

Diagonalizable

If an n x n matrix A is similar to a diagonal matrix, then A is said to be _____
(A = P^-1 D P, where D is a diagonal matrix and P is invertible)

Theorem 5.5: The Diagonalization Theorem

An n x n matrix A is diagonalizable if and only if A has n linearly independent eigenvectors. In fact, A = P D P^-1 if and only if the columns of P are n linearly independent eigenvector of A, and D is a diagonal matrix with diagonal entries are eigenvalues of A that corresponds to the eigenvectors in P

Theorem 5.6

An n x n matrix with n distinct eigenvalues is diagonalizable

Proof:

Theorem 5.2 → n distinct eigenvalues are corresponding to n linearly independent eigenvectors

Theorem 5.5 → the matric is diagonalizable

Cases of 3 × 3 Matrix Diagonalization (with real eigenvalues)

Three cases, check notes!!