Find Eigenvalues and Eigenvectors Quiz and Flashcards

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Questions and Answers

What is the condition to find an eigenvalue and an associated eigenvector of an n × n matrix A?

Av = 0
Av = λ + v
Av = vλ
Av = λv (correct)

What does the determinant of A - λI being equal to zero signify?

The matrix A is singular
The system has no solution
The system has a nontrivial solution (correct)
The matrix A has no eigenvalues

What is the characteristic polynomial of a matrix A?

p(λ) = det(A) + det(λI)
p(λ) = det(A + λI)
p(λ) = det(A - λI) (correct)
p(λ) = det(A)

What does the equation p(λ) = 0 represent?

The characteristic equation (D) Signup and view all the answers

In the context provided, if a root λ is found for the characteristic polynomial, what is it called?

Eigenvalue (D) Signup and view all the answers

When a nonzero column vector v satisfies Av = λv, what is it termed in relation to the matrix A?

Eigenvector (D) Signup and view all the answers

What is the characteristic polynomial of the given matrix?

$(−1 - λ)(4 - λ)(2 - λ)$ (D) Signup and view all the answers

Which of the following are roots of the characteristic polynomial?

$4$ (A) Signup and view all the answers

When finding eigenvectors, for which eigenvalue do we solve the homogeneous system?

$4$ (D) Signup and view all the answers

In the context of the matrix A, what does λ represent in the characteristic polynomial?

Eigenvalue (D) Signup and view all the answers

What is the value of λ2, one of the roots of the characteristic polynomial?

$-1$ (D) Signup and view all the answers

What is the determinant used to find the characteristic polynomial?

$det(A - λI)$ (C) Signup and view all the answers

What is the condition for a matrix to be singular?

Having at least one eigenvalue equal to 0 (C) Signup and view all the answers

What is the definition of diagonalization?

Transforming a matrix into a diagonal matrix of eigenvalues using its eigenvectors (C) Signup and view all the answers

How many linearly independent eigenvectors do we have when choosing z = 0?

Two (A) Signup and view all the answers

When λ = 3, what is the eigenvector left as an exercise?

[1, -1] (C) Signup and view all the answers

What is an eigenspace?

A subspace formed by eigenvectors corresponding to λ (D) Signup and view all the answers

For an n × n matrix A, how does the determinant relate to eigenvalues?

$\text{det} A = \prod_{i=1}^{n} \lambda_i$ (B) Signup and view all the answers

What is the value of y in the system of equations after Gaussian elimination?

0 (D) Signup and view all the answers

What is the value of z in the system of equations after Gaussian elimination?

0 (C) Signup and view all the answers

What is the value of x in the eigenvector obtained by choosing x = 1?

1 (C) Signup and view all the answers

What is the value of λ in the homogeneous system (A − λI )⃗ v = 0?

-1 (A) Signup and view all the answers

What is the relationship between y and z in the system of equations after Gaussian elimination and interchange of rows?

y = 3z/2 (A) Signup and view all the answers

What is the value of z in the general eigenvector         x −3z −3 −3  y  = 3z/2 = z 3/2 = 3/2 z z 1 1 (43)?

1 (B) Signup and view all the answers

What is the value of the eigenvalue λ in the given example?

λ = 2 (B) Signup and view all the answers

What is the general form of the eigenvector for the given matrix A?

x = −3z/2, y = 0, z arbitrary (B) Signup and view all the answers

What is the characteristic polynomial of the given matrix A?

det(A − λI ) = (λ + 1)(λ − 1)2 (A) Signup and view all the answers

What is the eigenvector corresponding to the eigenvalue λ = 1 in the given example?

      x −3/2 y  =  0  z  0  z (B) Signup and view all the answers

What is the determinant of the matrix (A − λI ) in the given example?

det(A − λI ) = (λ + 1)(λ − 1)2 (D) Signup and view all the answers

What is the matrix A in the given example?

A =   4 8 3  0   −1 − 2 0  0 −2 2 (C) Signup and view all the answers

Study Notes

Finding Eigenvalues and Eigenvectors

To find eigenvalues and eigenvectors, we need to solve the homogeneous system (A - λI)v = 0, which is equivalent to det(A - λI) = 0.
The determinant of A - λI is a polynomial of degree n, so we need to find the roots of the polynomial p(λ) = det(A - λI).

Characteristic Polynomial and Equation

The polynomial p(λ) = det(A - λI) is called the characteristic polynomial of A.
The equation p(λ) = 0 is termed the characteristic equation.
If λ is a root of p, it is termed an eigenvalue of A.

Eigenvectors

If v is a nonzero column vector satisfying Av = λv, it is an eigenvector of A.
We say that v is an eigenvector corresponding to the eigenvalue λ.

Example 1

Let A = [[2, -0.4707], [0.7481, 1.7481]], and v = [[0.2898], [0.9571]].
Then, Av = 2v, so v is an eigenvector of A corresponding to eigenvalue 2.

Example 2

Find the eigenvalues and eigenvectors of the matrix A = [[4, 8, 3], [0, -1, 0], [0, -2, 2]].
The characteristic polynomial is p(λ) = (-1 - λ)(4 - λ)(2 - λ) = 0.
The roots of the characteristic polynomial are λ1 = 4, λ2 = -1, and λ3 = 2.

Eigenvectors for λ = 4

We solve the homogeneous system (A - λI)v = 0 to find the eigenvectors.
The eigenvectors are [[-1, 1, 0]^T] and [[-1, 0, 1]^T].

Eigenvectors for λ = -1

We solve the homogeneous system (A - λI)v = 0 to find the eigenvectors.
The eigenvectors are [[-3z/2, 3z/2, z]^T], where z is a nonzero scalar.

Eigenvectors for λ = 2

We solve the homogeneous system (A - λI)v = 0 to find the eigenvectors.
The eigenvectors are [[-3z/2, 0, z]^T], where z is a nonzero scalar.

The Case of Repeated Roots

Some Properties of Eigenvalues and Eigenvectors:
- A matrix A is singular if and only if it has a 0 eigenvalue.
- The eigenvectors corresponding to λ form a subspace called an eigenspace.
- If A is an n × n matrix, then det A = ∏λi, i = 1 to n.

Diagonalization

Diagonalization involves using the eigenvectors of a matrix A to transform A into a diagonal matrix of eigenvalues.

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Description

Learn about the definition of eigenvalues and eigenvectors in linear algebra, and how to find them using the homogeneous system. Understand the conditions for a matrix to have eigenvalues and associated eigenvectors.

Eigenvalues and Eigenvectors: Finding Eigenvalues and Eigenvectors

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