Matrix Diagonalization and Exercises

Diagonalization of Matrices

• Although the power method only estimates the dominant eigenvalue of a matrix A, variations exist for estimating any eigenvalues of A starting from a good approximation A₀ of A.
• An example is the inverse power method.

Exercises on Matrices

• True or False Questions
  • Equivalent matrices are congruent if the matrices relating them are transposed.
  • 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 v corresponding to a non-zero eigenvalue λ, then v may not be an eigenvector of A⁻¹.
  • Eigenvectors with different eigenvalues can be linearly dependent or have a linear combination.
  • If an eigenvalue λ of a matrix A has algebraic multiplicity 2, then the matrix must have 2 linearly independent 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 of A, then the eigenspace L₁ of eigenvalue λ satisfies L₁=Nuc(f-λId) and dim L₁ = n-rank(A-λI) where Id is the identity transformation.
  • A non-diagonalizable 4 x 4 matrix with distinct eigenvalues λ₁, λ₂ can have geometric multiplicity d₁ = 2, a₂ = 3.
  • If an 8 x 8 matrix has only one eigenvalue λ ≠ 1 with geometric multiplicity 7, then the Jordan form of A has only one 1.

Applications Linéales et Matrices

• A system of linearly independent vectors that are transformed through a linear application may or may not remain linearly independent, depending on the application itself.
• The same matrix A can correspond to two different linear applications (f and g) for different bases.
• A matrix A defines a linear application independent of the bases chosen in the initial and final spaces.
• The subspace image of a linear application f has a dimension equal to the rank of any associated matrix.
• The null space Nuc(f) of a linear application f has a dimension equal to the difference between the total number of columns and the rank of any associated matrix.
• Every linear application is an endomorphism.
• Matrix change bases are non singular.
• Change of base matrices are the identity matrix.
• The matrix associated with a linear application does not change when the initial and final bases are changed.

Espacios Vectoriales

• A vector space is a direct sum of subspaces generated by the vectors in any basis.
• True or False Questions:
  • If for v, w ∈ R² with b > 0 such that is defined an internal operation (a, b) * (c, d) = (ac, b+d), then the operation has a neutral element.
  • Two algebraic structures obtained from sets A (with two different operations '+' and '*') have the same properties if they have the same operations.
  • The set formed by the zero vector of a vector space is always a subspace.
  • A subset U of a vector space V is a subspace of V.
  • (2, 2) ⊂ R² = {(11, 22) ∈ R² : 11 = 22} is a non-commutative group.
  • The number of bases of a vector space depends on its dimension.
  • A three-dimensional vector space cannot have a subspace of dimension greater than 3.
  • The sum of two subspaces is always a direct sum of subspaces.
  • The dimension of the sum of two subspaces is the sum of their dimensions.
  • The sum of two subspaces is always contained within the union of both subspaces.

Herramientas

• True or False Questions:
  • Each elementary row operation on a matrix corresponds to a unique elementary matrix.
  • The determinant of an n x n matrix is the 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 zero
  • A homogeneous system has a unique solution if and only if the rank of the coefficient matrix and the number of unknowns are the same.
  • A system of five equations with four variables and a coefficient matrix with rank 4 has a unique solution.
  • A system of five equations with four variables and a coefficient matrix with rank 4 might be inconsistent.
  • Numerical methods applied to consistent systems always find the exact solution.
  • The Gauss method transforms the augmented matrix into row echelon form.
  • The LU factorization method allows certain row operations (F→AF).

This quiz covers the concepts related to the diagonalization of matrices, including eigenvalues and eigenvectors. It includes true or false questions that test your understanding of the important properties and criteria related to matrices. Sharpen your skills and deepen your knowledge in linear algebra with this interactive set of questions.