# MTH603 Quiz 3 Solution and Discussion

• The Power method can be used only to find the eigenvalue of A that is largest in absolute value—we call this eigenvalue the dominant eigenvalue of A.

True
False

As presented here, the method can be used only to find the eigenvalue of A that is largest in absolute value—this eigenvalue is called the dominant … The eigenvectors corresponding to are called dominant eigenvectors of A. 1 i. 2, . . . , n.

• Central difference method seems to be giving a better approximation, however it requires more computations.

True
False

Numerical method. In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm.

Reff

• If x is an eigen value corresponding to eigen value of V of a matrix A. If a is any constant, then x – a is an eigen value corresponding to eigen vector V is an of the matrix A - a I.

True
False

If an eigenvalue l of A is known, the corresponding eigenvector(s) may be obtained by … l of a matrix A is the maximum number of linearly independent eigen vectors x of A … If v1, v2, …, vn are the eigenvectors associated with the respective … the eigenvalues of A and then if some of them are multiple, to check if there exist …

• If n x n matrices A and B are similar, then they have the different eigenvalues (with the same multiplicities).

True
False

If A and B are positive definite, is A + B positive definite? We don’t know … to A. If two matrices have the same n distinct eigenvalues, they’ll be similar to the same diagonal

• For differences methods we require the set of values.

True
False

Method of Differences. The method of finite differences gives us a way to calculate a polynomial using its values at several consecutive points. This is often a good approach to finding the general term in a pattern, if we suspect that it follows a polynomial form.

• The characteristics polynomial of a 3x 3 identity matrix is __________, if x is the eigen values of the given 3 x 3 identity matrix. where symbol ^ shows power.

• (x-1)^3
• (x+1)^3
• x^3-1
• x^3+1

• The Jacobi’s method is a method of solving a matrix equation on a matrix that has no zeros along its main diagonal.

True
False

Jacobi Method.
The Jacobi method is a method of solving a matrix equation on a matrix that has no zeros along its main diagonal (Bronshtein and Semendyayev 1997, p. … Each diagonal element is solved for, and an approximate value plugged in. The process is then iterated until it converges.

• The Power method can be used only to find the eigenvalue of A that is largest in absolute value—we call this eigenvalue the dominant eigenvalue of A

True
False

For large values of n, polynomial equations like this one are difficult and time-consuming to solve. Moreover, numerical techniques for approximating roots of polynomial equations of high degree are sensitive to rounding errors. In this section we look at an alternative method for approximating eigenvalues. As presented here, the method can be used only to find the eigenvalue of A that is largest in absolute value—we call this eigenvalue the dominant eigenvalue of A. Although this restriction may seem severe, dominant eigenval- ues are of primary interest in many physical applications.

• Eigenvalues of a symmetric matrix are all _________.

• real
• zero
• positive
• negative

In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, … geometry, for each tangent space to a manifold may be endowed with an inner product, giving rise to what is called a Riemannian manifold. … Every real symmetric matrix is Hermitian, and therefore all its eigenvalues are real.

• The determinant of a _______ matrix is the product of the diagonal elements.

• diagonal
• upper triangular
• lower triangular
• scalar

By linearity then, the determinant of AB is the product of the diagonal elements of A times the determinant of B, that is, it is the product of the determinant of A and that of B, as we have claimed. and we will have det A = det A’ , and det AB = det A’B.

• The Gauss-Seidel method is applicable to strictly diagonally dominant or symmetric positive definite matrices A.

True
False

Explanation: Gauss-Seidel method is applicable to strictly diagonally dominant or symmetric positive definite matrices because only in this case convergence is possible. 7. Gauss seidal requires less number of iterations than Jacobi’s method. … Gauss-seidal is used for solving system of linear equations.

• The Jacobi’s method is a method of solving a matrix equation on a matrix that has ____ zeros along its main diagonal.

Unity
Zero

Reff

• If n x n matrices A and B are similar, then they have the different eigenvalues (with the same multiplicities).

True
False
A proof of the fact that similar matrices have the same eigenvalues and their … Show that if A and B are similar matrices, then they have the same eigenvalues and their … Eigenvalues and their Algebraic Multiplicities of a Matrix with a Variable … Suppose that all the eigenvalues of A are distinct and the matrices A and B …

• A 3 x 3 identity matrix have three and different eigen values.
True
False

Eigenvalues and Eigenvectors of a 3 by 3 matrix. … The picture is more complicated, but as in the 2 by 2 case, our best insights come from finding the matrix’s eigenvectors: that is, those vectors whose direction the transformation leaves unchanged.

• Power method is applicable if the eigen vectors corresponding to eigen values are linearly independent.

Power method is applicable if the eigenvectors corresponding to eigenvalues are linearly independent.

True
False

linearly independent eigenvectors. … We could still compute the ratio of corresponding components for some index, but the … Using your power method code, try to determine the largest eigenvalue of the … So far, the methods we have discussed seem suitable for finding one or a few eigenvalues and eigenvectors at a time.

Reff

• The determinant of a diagonal matrix is the product of the diagonal elements.
True

The determinant of a lower triangular matrix (or an upper triangular matrix) is the product of the diagonal entries. In particular, the determinant of a diagonal matrix is the product of the diagonal entries.

Reff

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