Applied Linear Algebra: Eigenvalues and Eigenvectors

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Questions and Answers

For a 3x3 matrix, which statement about eigenvalues is always true?

  • The sum of the eigenvalues is equal to the trace of the matrix. (correct)
  • The product of the eigenvalues is equal to the trace of the matrix.
  • The sum of the eigenvalues is equal to the determinant of the matrix.
  • The eigenvalues are always distinct.

The geometric multiplicity of an eigenvalue can be greater than its algebraic multiplicity.

False (B)

What is the significance of verifying the Cayley-Hamilton theorem for a matrix?

Allows computation of the inverse of a matrix.

If two matrices are similar, they have the same ________.

<p>eigenvalues</p>
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Match the following terms related to quadratic forms with their descriptions:

<p>Rank = Number of non-zero eigenvalues Index = Number of positive eigenvalues Signature = Difference between the number of positive and negative eigenvalues Nature = Value class of a quadratic form</p>
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Which of the following is a direct application of eigenvalue decomposition?

<p>Data compression using Principal Component Analysis (PCA) (B)</p>
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A singular value of a matrix can be negative.

<p>False (B)</p>
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In the context of quadratic forms, what does the 'index' signify?

<p>The number of positive eigenvalues.</p>
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The Cayley-Hamilton theorem states that every square matrix satisfies its own ________ equation.

<p>characteristic</p>
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Match the matrix type with its property regarding eigenvalues:

<p>Symmetric Matrix = All eigenvalues are real Orthogonal Matrix = All eigenvalues have absolute value 1 Positive Definite Matrix = All eigenvalues are positive Skew-Symmetric Matrix = Eigenvalues are either zero or purely imaginary</p>
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If a matrix A has an eigenvalue of 0, what can be concluded?

<p>A is singular (D)</p>
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For any matrix A, the singular values of A are the same as the eigenvalues of A.

<p>False (B)</p>
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What is the purpose of finding a modal matrix?

<p>Diagonalize a matrix</p>
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The algebraic multiplicity of an eigenvalue is the number of times it appears as a ________ of the characteristic polynomial.

<p>root</p>
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Match the quadratic form characteristic with its mathematical condition:

<p>Positive definite = All eigenvalues &gt; 0 Negative definite = All eigenvalues &lt; 0 Indefinite = Has both positive and negative eigenvalues Positive semidefinite = All eigenvalues ≥ 0</p>
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Which of the following statements is true regarding eigenvectors?

<p>Eigenvectors corresponding to distinct eigenvalues are always linearly independent. (A)</p>
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If a matrix is diagonalizable, it must have n linearly independent eigenvectors, where n is the size of the matrix.

<p>True (A)</p>
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What is a key difference between eigenvalue decomposition and singular value decomposition (SVD)?

<p>SVD works on non-square matrices</p>
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For a positive definite matrix, all of its principal ________ are positive.

<p>minors</p>
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In the context of linear algebra, match the following concepts with their descriptions:

<p>Eigenvalue = A scalar associated with a linear transformation that describes how a vector changes in magnitude. Eigenvector = A vector that, when acted upon by a linear transformation, only changes by a scalar factor. Eigenspace = The set of all eigenvectors associated with a particular eigenvalue, together with the zero vector. Characteristic polynomial = A polynomial whose roots are the eigenvalues of the matrix.</p>
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Flashcards

What is an eigenvalue?

A scalar λ such that det(A - λI) = 0, where A is a matrix and I is the identity matrix.

What is an eigenvector?

A non-zero vector v such that Av = λv, where A is a matrix and λ is an eigenvalue.

What is Algebraic Multiplicity?

The power of (λ - λi) in the characteristic polynomial, where λi is an eigenvalue.

What is Geometric Multiplicity?

The dimension of the eigenspace corresponding to an eigenvalue λ.

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What is the Cayley-Hamilton Theorem?

Every square matrix satisfies its own characteristic equation.

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What does it mean for a matrix to be similar to a diagonal matrix?

A matrix A is similar to a diagonal matrix D if there exists an invertible matrix P such that A = PDP^(-1).

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What is a Quadratic Form?

Representation of a polynomial with terms all of degree two.

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What is matrix notation for quadratic forms?

A matrix representing a quadratic form.

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What is the nature (value class) of a quadratic form?

Classification by positive, negative, or indefinite values.

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What is the rank of a quadratic form?

The number of non-zero eigenvalues of the matrix.

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What is the index of a quadratic form?

The number of positive eigenvalues of the matrix.

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What is the signature of a quadratic form?

The difference between the number of positive and negative eigenvalues.

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What is Singular Value Decomposition (SVD)?

Decomposing a matrix into U, Σ, and VT, where U and V are orthogonal matrices, and Σ is a diagonal matrix with singular values.

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Study Notes

Applied Linear Algebra - Assignment II, Unit III: Eigenvalues and Eigenvectors

  • Find the eigenvalues and eigenvectors for the following matrices:
    • Matrix (i): 
      | 1  0 -17 |
| 1  2  1  |
| 2  2  3  |
    • Matrix (ii): 
      |  8 -6  2 |
| -6  7 -4 |
|  2 -4  3 |
  • Determine algebraic and geometric multiplicity of the following matrices:
    • Matrix (i): 
      | 0  1  0 |
| 0  0  1 |
| 1 -3  3 |
    • Matrix (ii): 
      | 2  1  1 |
| 1  2  1 |
| 0  0  1 |
  • Verify Cayley-Hamilton theorem for the following matrix and find A⁻¹:
    • Matrix A: 
      | 0  1  0 |
| 0  0  1 |
| 2 -5  4 |
  • Show that the following matrices are similar to diagonal matrices and find the diagonal and modal matrix in each case:
    • Matrix (i): 
      |  4  2 -2 |
| -5  3  2 |
| -2  4  1 |
    • Matrix (ii): 
      |  -9  4  4 |
|  -8  3  4 |
| -16  8  7 |
  • Express the following quadratic forms in matrix notation:
    • Quadratic form (i): 2x² + 3y² - 5z² - 2xy + 6xz - 10yz
    • Quadratic form (ii): x₁² + 2x₂² + 3x₃² + x₄² - 2x₁x₂ + 4x₁x₃ - 2x₁x₄ + 4x₂x₃ - 6x₂x₄ - 8x₃x₄
  • Write down the quadratic forms corresponding to the following matrices:
    • Matrix (i): 
      | 2  1  5 |
| 1  3 -2 |
| 5 -2  4 |
    • Matrix (ii): 
      | 0  1  2  3 |
| 1  2  3  4 |
| 2  3  4  5 |
| 3  4  5  6 |
  • Determine the nature (value class), rank, index, and signature of the following quadratic forms:
    • Quadratic form (i): 6x₁² + 3x₂² + 3x₃² - 4x₁x₂ - 2x₂x₃ + 4x₃x₁
    • Quadratic form (ii): -3x₁² - 3x₂² - 3x₃² - 2x₁x₂ - 2x₁x₃ + 2x₂x₃
  • Find a singular value of the matrix:
    • Matrix (i): 
      | 1 -1 |
| 1  1 |
    • Matrix (ii): 
      | -3  0 |
|  0 -4 |
  • Find a singular value decomposition of the matrix:
    • Matrix (i): 
      | 1  1 |
| 0  1 |
| 1  0 |
    • Matrix (ii): 
      | -2 -1  2 |
| -1  2 -2 |

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