Diagonalization of Matrices Quiz

Diagonalization of Matrices

• Although the power method only estimates the dominant eigenvalue of a matrix A, there are variations that estimate any eigenvalue of A based on a good approximation A₀ of A.
• Example: The inverse power method

Exercises for Self-Evaluation

• Determine if the following statements are true
  • Two equivalent matrices are congruent if the matrices that relate them are transposes.
  • The operation P⁻¹AP can only be performed if A is square.
  • The zero vector (0) cannot be an eigenvector.
  • If A is a regular matrix with eigenvector v associated with a non-zero eigenvalue λ, then v may not be an eigenvector of A⁻¹.
  • Eigenvectors corresponding to distinct eigenvalues can be linearly dependent (not linearly independent).
  • If an eigenvalue λ of a matrix A has an algebraic multiplicity of 2, then the matrix must have 2 eigenvectors that form a basis for the subspace of eigenvectors associated with λ.
  • Similar matrices have the same characteristic roots, and a linear transformation is diagonalizable if there exists a basis of eigenvectors.

Linear Applications and Matrices

• If a system of vectors is linearly independent, then the images of those vectors under a linear transformation are also linearly independent, or linearly dependent depending on the transformation. This is not always the case.
• The same matrix A can represent different linear transformations depending on the bases used.
• A matrix A determines a linear transformation that is independent of the bases in the initial and final vector spaces.
• The subspace of images of a linear transformation has a dimension equal to the rank of any matrix associated with it.
• The dimension of the kernel of a linear transformation is related to the dimension of the associated matrix. (e.g., Dim Nuc(f) ≤ m; if the matrix A associated with f is n × m)
• All linear transformations are endomorphisms ? (The question mark indicates this is not necessarily true)
• Change of basis matrices are not singular.
• Linear transformations associated with a change of basis are the identity transformation.
• The matrix associated with a linear transformation does not change with different bases in the initial and final spaces.

Vector Spaces

• A vector space is a direct sum of subspaces generated by the vectors in any of its bases.
• Determining if the following statements are true:
  • If a binary operation * is defined on a set V = {(a, b) ∈ R²: b > 0}, where (a, b)*(c, d) = (ac, b + d), then V has an identity element.
  • If two different operations "+" and "*" are defined on a set A, the resulting algebraic structures (A, +) and (A, *) are necessarily the same? (Likely not.)
  • The set containing only the zero vector (0) in a vector space V is always a subspace of V.
  • A subset U of a vector space V can be a vector space with different operations, but it is not guaranteed that it is a subspace of V.
  • The set U = {(x₁, x₂) ∈ R²: x₁ = x₂} is not an abelian group.
  • The number of bases of a vector space depends on its dimension.
  • A 3-dimensional vector space cannot have subspaces with dimension greater than 3.
  • The sum of two subspaces is not always a direct sum.
  • The dimension of the sum of two subspaces is equal to the sum of their dimensions.
  • The sum of two subspaces is always contained within the union of the two subspaces.

Tools/Matrices

• Determining if statements about matrices and operations on matrices are true.
  • Each elementary row operation has a corresponding unique elementary matrix.
  • The determinant of an n x n matrix is a sum of n! terms.
  • An inverse of a diagonal matrix, when it exists, is also diagonal.
  • If the determinant of an n x n matrix is zero, then the rank is less than n.
  • A homogeneous system is consistent and determined if, and only if, the rank of the coefficient matrix equals the number of unknowns.
  • A system with 5 equations and 4 unknowns, where the rank of the augmented matrix is 4, has a unique solution.
  • A system with 5 equations, 4 unknowns, and a rank-4 coefficient matrix can be inconsistent.
  • Numerical methods for solving compatible systems always yield the exact solution.
  • The Gaussian method transforms an augmented matrix into a row-reduced echelon form.
  • Row operations of the type F → AF (where A ≠ 0) are permissible in LU factorization.

Test your understanding of matrix diagonalization concepts, including eigenvalues and eigenvectors. This quiz covers true/false statements related to matrix properties and the inverse power method. Challenge yourself with these self-evaluation exercises to deepen your knowledge of linear algebra.