Diagonalization of Matrices Quiz

Diagonalization of Matrices

• Although the power method only estimates the dominant eigenvalue of a matrix A, variations exist to estimate any eigenvalue of A from an approximate value A₀. One example is the inverse power method.

Exercises in Self-Evaluation

• Determine if the following statements are true:
  • Two equivalent matrices are congruent if the matrices relating them are transposes.
  • The operation P⁻¹AP can only be performed if A is a square matrix.
  • The zero vector cannot be an eigenvector.
  • If A is a regular matrix with an eigenvector associated with a non-zero eigenvalue λ, then v may not be an eigenvector of A⁻¹.
  • Eigenvectors corresponding to different eigenvalues can be linearly dependent.
  • 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 eigenspace associated with λ.
  • The characteristic roots of similar matrices are equal, and a linear transformation is diagonalizable if a basis of eigenvectors exists.
  • If A is an n x n matrix associated with a linear transformation f and λ is an eigenvalue, then the eigenspace Lλ satisfies Lλ = Nuc(f - λId) and dim(Lλ)= n - rank(A - λI) where Id is the identity linear transformation.
  • If A is a non-diagonalizable 4 x 4 matrix with distinct eigenvalues λ₁ and λ₂, it can occur that dim(E_λ₁) = 2 and dim(E_λ₂) = 3 .
  • If A is an 8 x 8 matrix with only one eigenvalue λ ≠ 1 and a geometric multiplicity of 7, the Jordan matrix of A contains only one 1.

Linear Applications and Matrices

• Determine if the following statements are true:
  • For a function f: V → W, where V and W are vector spaces, it is sufficient for f(x + y) = f(x) + f(y) for all x, y ∈ V to be linear.
  • If a set of vectors is linearly independent, the set of images of the vectors under a linear transformation will be linearly independent or dependent.
  • The same matrix A can correspond to different linear transformations f and g if different bases are used.
  • A matrix A determines a linear transformation that does not depend on the chosen bases in the initial and final spaces.
  • The image subspace of a linear transformation f has a dimension equal to the rank of any associated matrix.
  • The kernel subspace Nuc(f) of a linear transformation f satisfies dim(Nuc(f)) = m - rank(A), where A is any m x n matrix associated with f.
  • All linear transformations are endomorphisms.
  • Change of basis matrices are non-singular.
  • Linear transformations associated with a change of bases are the identity transformation.
  • The matrix associated with a linear transformation remains unchanged if the basis of the initial and final spaces are changed.

Vector Spaces

• Determine if the following statements are true:
  • If an inner operation (a, b) * (c, d) = (ac, b + d) is defined on V = {(a, b) ∈ R² : b > 0}, then (V, *) has an identity element.
  • If two operations "+" and "*" are defined on a set A, the resulting algebraic structures (A, +) and (A, *) are necessarily identical.
  • The set containing only the zero vector {0} is a subspace of a vector space.
  • A subset U of a vector space V, which is also a vector space under its own operations, 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 of dimension greater than 3.
  • The sum of two subspaces is 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 their union.

Tools

• Determine if the following statements are true:
  • Each elementary row operation corresponds to a unique elementary matrix.
  • The definition of the determinant of an n x n matrix involves a sum of n! terms.
  • The inverse of a diagonal matrix, if it exists, is also diagonal.
  • If the determinant of an n x n matrix is zero, then its rank is n.
  • A homogeneous system is consistent and uniquely solvable if and only if the rank of the coefficient matrix equals the number of unknowns.
  • A system of 5 equations with a 4 x 4 coefficient matrix and 4 unknowns has a unique solution.
  • A system of 5 equations with a 4 x 4 coefficient matrix and 4 unknowns can be inconsistent.
  • Numerical methods always yield an exact solution for consistent systems of equations.
  • The Gauss method transforms the augmented matrix into row-echelon form.
  • Row operations of the type Fᵢ → αFᵢ (α ≠ 0) are allowed in the LU factorization method.

Test your understanding of matrix diagonalization concepts, including eigenvalues and eigenvectors. This quiz covers the properties of matrices and their transformations, challenging you to evaluate statements critically. Ideal for students studying linear algebra.