Eigenvalues and Eigenvectors: Finding Eigenvalues and Eigenvectors
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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?

    <p>The characteristic equation</p> Signup and view all the answers

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

    <p>Eigenvalue</p> 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?

    <p>Eigenvector</p> Signup and view all the answers

    What is the characteristic polynomial of the given matrix?

    <p>$(−1 - λ)(4 - λ)(2 - λ)$</p> Signup and view all the answers

    Which of the following are roots of the characteristic polynomial?

    <p>$4$</p> Signup and view all the answers

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

    <p>$4$</p> Signup and view all the answers

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

    <p>Eigenvalue</p> Signup and view all the answers

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

    <p>$-1$</p> Signup and view all the answers

    What is the determinant used to find the characteristic polynomial?

    <p>$det(A - λI)$</p> Signup and view all the answers

    What is the condition for a matrix to be singular?

    <p>Having at least one eigenvalue equal to 0</p> Signup and view all the answers

    What is the definition of diagonalization?

    <p>Transforming a matrix into a diagonal matrix of eigenvalues using its eigenvectors</p> Signup and view all the answers

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

    <p>Two</p> Signup and view all the answers

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

    <p>[1, -1]</p> Signup and view all the answers

    What is an eigenspace?

    <p>A subspace formed by eigenvectors corresponding to λ</p> Signup and view all the answers

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

    <p>$\text{det} A = \prod_{i=1}^{n} \lambda_i$</p> Signup and view all the answers

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

    <p>0</p> Signup and view all the answers

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

    <p>0</p> Signup and view all the answers

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

    <p>1</p> Signup and view all the answers

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

    <p>-1</p> 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?

    <p>y = 3z/2</p> 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)?

    <p>1</p> Signup and view all the answers

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

    <p>λ = 2</p> Signup and view all the answers

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

    <p>x = −3z/2, y = 0, z arbitrary</p> Signup and view all the answers

    What is the characteristic polynomial of the given matrix A?

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

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

    <p>      x −3/2 y  =  0  z  0  z</p> Signup and view all the answers

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

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

    What is the matrix A in the given example?

    <p>A =   4 8 3  0   −1 − 2 0  0 −2 2</p> 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.

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