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Diagonalization and Eigenvalues

This quiz covers diagonalization concepts, eigenvalues, eigenvectors, characteristic polynomial, and related theorems in linear algebra.

10 Questions Published
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Questions

Question 1 Multiple Choice (Single Answer)

What is the process of finding a matrix that is similar to a given matrix called?

  1. Diagonalization
  2. Triangularization
  3. Orthogonalization
  4. Jordanization
Question 2 Multiple Choice (Single Answer)

What is an eigenvalue of a matrix?

  1. A scalar that, when multiplied by the matrix, produces the matrix itself
  2. A scalar that, when multiplied by the matrix, produces the zero matrix
  3. A scalar that, when multiplied by the matrix, produces a diagonal matrix
  4. A scalar that, when multiplied by the matrix, produces an orthogonal matrix
Question 3 Multiple Choice (Single Answer)

What is an eigenvector of a matrix?

  1. A vector that, when multiplied by the matrix, produces the eigenvalue of the matrix
  2. A vector that, when multiplied by the matrix, produces the zero vector
  3. A vector that, when multiplied by the matrix, produces a diagonal matrix
  4. A vector that, when multiplied by the matrix, produces an orthogonal matrix
Question 4 Multiple Choice (Single Answer)

What is the diagonalization theorem?

  1. A theorem that states that every square matrix can be diagonalized
  2. A theorem that states that every square matrix has at least one eigenvalue
  3. A theorem that states that every square matrix has at least one eigenvector
  4. A theorem that states that every square matrix is similar to a diagonal matrix
Question 5 Multiple Choice (Single Answer)

What is the Jordan canonical form of a matrix?

  1. A matrix that is similar to a given matrix and has all of its eigenvalues on the diagonal
  2. A matrix that is similar to a given matrix and has all of its eigenvectors as its columns
  3. A matrix that is similar to a given matrix and has all of its eigenvalues on the diagonal and all of its eigenvectors as its columns
  4. A matrix that is similar to a given matrix and has all of its eigenvalues on the diagonal and all of its eigenvectors as its rows
Question 6 Multiple Choice (Single Answer)

What is the characteristic polynomial of a matrix?

  1. A polynomial whose roots are the eigenvalues of the matrix
  2. A polynomial whose roots are the eigenvectors of the matrix
  3. A polynomial whose roots are the diagonal elements of the matrix
  4. A polynomial whose roots are the off-diagonal elements of the matrix
Question 7 Multiple Choice (Single Answer)

What is the minimal polynomial of a matrix?

  1. A polynomial that is the lowest-degree polynomial that annihilates the matrix
  2. A polynomial that is the lowest-degree polynomial that has the matrix as a root
  3. A polynomial that is the lowest-degree polynomial that has the eigenvalues of the matrix as its roots
  4. A polynomial that is the lowest-degree polynomial that has the eigenvectors of the matrix as its roots
Question 8 Multiple Choice (Single Answer)

What is the Cayley-Hamilton theorem?

  1. A theorem that states that every square matrix satisfies its own characteristic polynomial
  2. A theorem that states that every square matrix satisfies its own minimal polynomial
  3. A theorem that states that every square matrix is similar to a diagonal matrix
  4. A theorem that states that every square matrix has at least one eigenvalue
Question 9 Multiple Choice (Single Answer)

What is the eigenvalue-eigenvector method for solving a system of linear differential equations?

  1. A method for solving a system of linear differential equations by finding the eigenvalues and eigenvectors of the coefficient matrix
  2. A method for solving a system of linear differential equations by finding the characteristic polynomial of the coefficient matrix
  3. A method for solving a system of linear differential equations by finding the minimal polynomial of the coefficient matrix
  4. A method for solving a system of linear differential equations by finding the Cayley-Hamilton theorem of the coefficient matrix
Question 10 Multiple Choice (Single Answer)

What is the power method for finding the largest eigenvalue and corresponding eigenvector of a matrix?

  1. A method for finding the largest eigenvalue and corresponding eigenvector of a matrix by repeatedly multiplying the matrix by a random vector
  2. A method for finding the largest eigenvalue and corresponding eigenvector of a matrix by repeatedly multiplying the matrix by a vector that is orthogonal to the previous vector
  3. A method for finding the largest eigenvalue and corresponding eigenvector of a matrix by repeatedly multiplying the matrix by a vector that is parallel to the previous vector
  4. A method for finding the largest eigenvalue and corresponding eigenvector of a matrix by repeatedly multiplying the matrix by a vector that is equal to the previous vector
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