Diagonalization of Matrices Quiz

Diagonalization of Matrices

• Although the power method only estimates (if it exists) the strictly dominant eigenvalue of a matrix A, there are various variations of it that allow estimation of any eigenvalue of A starting from a good approximation A₀ of A. For example, the inverse power method.

Exercises for Self-Assessment

• 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 cannot be an eigenvector.
  • If A is a regular matrix with an eigenvector associated to a nonzero 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 algebraic multiplicity 2, then the matrix must have 2 eigenvectors that form a basis for the eigenspace associated with the eigenvalue λ.
  • It is true that the characteristic roots of two similar matrices are equal, and also, an endomorphism is diagonalizable if there is a basis of eigenvectors.
  • If A is an n×n matrix associated with an endomorphism f and λ is an eigenvalue of it, then the eigenspace Lλ of eigenvalue λ satisfies: Lλ = Ker(f - λId) and Dim Lλ = n - rank(A - λId) where Id is the identity linear application.
  • If A is a non-diagonalizable 4×4 matrix with two distinct eigenvalues λ₁ and λ₂, it can happen that dim(Eλ₁) = 2 and dim(Eλ₂) = 3.
  • If A is an 8×8 matrix and only has one eigenvalue λ ≠ 1 with geometric multiplicity 7, then the Jordan matrix of A has only one 1.

Linear Applications and Matrices

• Determine if the following statements are true:
  • For an application 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 system of vectors is linearly independent, the system formed by their images under a linear application will be linearly independent or linearly dependent depending on the given application.
  • The same matrix A can correspond to two different linear applications f and g for different bases.
  • A matrix A determines a linear application that does not depend on the bases chosen in the initial and final spaces.
  • The image subspace of a linear application f has a dimension equal to the rank of any associated matrix.
  • The kernel subspace Ker(f) of a linear application f satisfies Dim Ker(f) = m - rank(A), where A is any m×n matrix associated with f.
  • All linear applications are endomorphisms.
  • Change-of-basis matrices are non-singular.
  • Linear applications associated with a change of basis are the identity linear application.
  • Even if the bases of the initial and final spaces of a linear application are changed, the matrix associated with the application does not change.

Vector Spaces

• Determine if the following statements are true:
  • If in the set V = {(a, b) ∈ R²: b > 0}, the internal operation (a, b)*(c, d) = (ac, b + d) is defined, then (V, *) has a neutral element.
  • If two different operations "+" and "*" are defined on a set A, the algebraic structures obtained in (A, +) and (A, *) necessarily coincide.
  • The set formed by the zero element {0} of a vector space V is always a subspace of V.
  • Even if a subset U of a vector space V is itself a vector space with different operations from those defined in V, we cannot say that U is a subspace of V.
  • The subset U = {(x₁, x₂) ∈ R²: x₁ = x₂} is not an abelian group.
  • The number of bases of a vector space depends on its dimension.
  • If a vector space has dimension 3, it cannot have subspaces with dimension greater than 3.
  • The sum of two subspaces is always a direct sum.
  • The dimension of the sum of two subspaces is always the sum of their dimensions.
  • The sum of two subspaces is always contained in their union.

Tools

• Determine if the following statements are true:
  • Each elementary row operation has a unique associated elementary matrix.
  • In the definition of the determinant of an n×n matrix, there is a sum of n! terms.
  • The inverse of a diagonal matrix, if it exists, is also diagonal.
  • If the determinant of an n×n matrix is zero, then its rank is n.
  • A homogeneous system is consistent and determined if and only if the rank of the coefficient matrix and the number of unknowns are equal.
  • If a system of linear equations has 5 equations, the rank of the augmented matrix is 4 and there are 4 unknowns, the system has a unique solution.
  • If a system of linear equations has 5 equations, the rank of the coefficient matrix is 4 and there are 4 unknowns, the system can be inconsistent.
  • Numerical methods applied to the solution of consistent systems of equations always provide the exact solution.
  • The Gauss method consists of transforming the augmented matrix into a reduced row-echelon form.
  • In the LU factorization method, row operations of the type F → AF, where A ≠ 0, are permitted.

Test your understanding of diagonalization and eigenvalues with this quiz. Explore concepts such as congruence, eigenvectors, and properties of matrices. Determine the truth of various statements related to matrix theory.