Matrix Diagonalization and Exercises

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Questions and Answers

Which of the following statements about the power method is TRUE?

It requires the matrix to be non-square for accurate estimations.

It uses the characteristic polynomial to find eigenvalues directly.

It can estimate all eigenvalues regardless of initial approximation.

It can only estimate the strictly dominant eigenvalue. (correct)

Which condition must a matrix A satisfy for the operation P⁻¹AP to be valid?

A must be singular.

A must be symmetric.

A must be square. (correct)

A must have distinct eigenvalues.

What can be said about eigenvectors corresponding to distinct eigenvalues?

They must be orthogonal in all cases.

They can be dependent on each other.

They are always linearly independent. (correct)

They can be linked together in a linear combination.

If a non-diagonalizable matrix has two distinct eigenvalues, what could their algebraic and geometric multiplicities be?

One eigenvalue can have a multiplicity of 2 and the other can have a multiplicity of 3. Signup and view all the answers

What conclusion can be drawn from a linear application f: V→W to be linear?

It suffices that f(x+y)=f(x)+f(y) for any x,y in V. Signup and view all the answers

What does the statement about eigenvalue λ with algebraic multiplicity 2 imply?

The eigenvalue must indicate diagonalizability. Signup and view all the answers

In the context of a matrix A being of order 8 with an eigenvalue λ ∉ {1}, what can be deduced?

There is only one Jordan block of size one. Signup and view all the answers

Which relation holds true about equivalent matrices and congruence?

Two equivalent matrices are congruent if the matrices that relate them are transposed. Signup and view all the answers

What is the relationship between the rank of a matrix and the dimension of the image subspace of a linear application?

The dimension of the image subspace is equal to the rank of the matrix. Signup and view all the answers

Which statement about the kernel subspace Nuc(f) of a linear application is incorrect?

Dim Nuc(f) is always equal to m minus the rank of A. Signup and view all the answers

Which assertion about base change matrices is true?

Base change matrices are singular. Signup and view all the answers

If a vector space V has dimension 3, which of the following is necessarily true?

No subspace can have dimension greater than 3. Signup and view all the answers

How does the dimension of the sum of two subspaces relate to their individual dimensions?

It is always less than the sum of their individual dimensions. Signup and view all the answers

What can be concluded about the set U formed by the neutral element of a vector space?

U lacks closure under vector addition. Signup and view all the answers

Regarding the properties of linear applications, what is true?

All linear applications are endomorphisms. Signup and view all the answers

Which statement about elementary row operations and their matrices is correct?

Each elementary row operation has a unique associated elementary matrix. Signup and view all the answers

Study Notes

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).

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Description

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.