30 Questions
What is the condition to find an eigenvalue and an associated eigenvector of an n × n matrix A?
Av = λv
What does the determinant of A - λI being equal to zero signify?
The system has a nontrivial solution
What is the characteristic polynomial of a matrix A?
p(λ) = det(A - λI)
What does the equation p(λ) = 0 represent?
The characteristic equation
In the context provided, if a root λ is found for the characteristic polynomial, what is it called?
Eigenvalue
When a nonzero column vector v satisfies Av = λv, what is it termed in relation to the matrix A?
Eigenvector
What is the characteristic polynomial of the given matrix?
$(−1 - λ)(4 - λ)(2 - λ)$
Which of the following are roots of the characteristic polynomial?
$4$
When finding eigenvectors, for which eigenvalue do we solve the homogeneous system?
$4$
In the context of the matrix A, what does λ represent in the characteristic polynomial?
Eigenvalue
What is the value of λ2, one of the roots of the characteristic polynomial?
$-1$
What is the determinant used to find the characteristic polynomial?
$det(A - λI)$
What is the condition for a matrix to be singular?
Having at least one eigenvalue equal to 0
What is the definition of diagonalization?
Transforming a matrix into a diagonal matrix of eigenvalues using its eigenvectors
How many linearly independent eigenvectors do we have when choosing z = 0?
Two
When λ = 3, what is the eigenvector left as an exercise?
[1, -1]
What is an eigenspace?
A subspace formed by eigenvectors corresponding to λ
For an n × n matrix A, how does the determinant relate to eigenvalues?
$\text{det} A = \prod_{i=1}^{n} \lambda_i$
What is the value of y in the system of equations after Gaussian elimination?
0
What is the value of z in the system of equations after Gaussian elimination?
0
What is the value of x in the eigenvector obtained by choosing x = 1?
1
What is the value of λ in the homogeneous system (A − λI )⃗ v = 0?
-1
What is the relationship between y and z in the system of equations after Gaussian elimination and interchange of rows?
y = 3z/2
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
What is the value of the eigenvalue λ in the given example?
λ = 2
What is the general form of the eigenvector for the given matrix A?
x = −3z/2, y = 0, z arbitrary
What is the characteristic polynomial of the given matrix A?
det(A − λI ) = (λ + 1)(λ − 1)2
What is the eigenvector corresponding to the eigenvalue λ = 1 in the given example?
x −3/2 y = 0 z 0 z
What is the determinant of the matrix (A − λI ) in the given example?
det(A − λI ) = (λ + 1)(λ − 1)2
What is the matrix A in the given example?
A = 4 8 3 0 −1 − 2 0 0 −2 2
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.
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.
Make Your Own Quizzes and Flashcards
Convert your notes into interactive study material.
Get started for free
