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

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

• If we wanted to find the value of a definite integral with an infinite limit, we can instead replace the infinite limit with a variable, and then take the limit as this variable goes to _________.

constant
finite
infinity
zero

We will replace the infinity with a variable (usually t ), do the integral and then take the limit of the result as t goes to infinity. On a side note, notice that the area under a curve on an infinite interval was not infinity as we might have suspected it to be. In fact, it was a surprisingly small number.

• Euler’s Method numerically computes the approximate derivative of a function.

False

What is Euler method used for?
In mathematics and computational science, the Euler method (also called forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value.

Reff

• An improper integral is the limit of a definite integral as an endpoint of the interval of integration approaches either a specified real number or 8 or -8 or, in some cases, as both endpoints approach limits.

True

• The Trapezoidal Rule is an improvement over using rectangles because we have much less “missing” from our calculations. We used ________ to model the curve in trapezoidal Rule.

• straight lines
• curves
• parabolas
• constant

Trapezoidal Rule, we used straight lines to model a curve and learned that it was an improvement over using rectangles for finding areas under curves because we had much less “missing” from each segment.

Reff

• In Runge – Kutta Method, we do not need to calculate higher order derivatives and find greater accuracy.

True

• The need of numerical integration arises for evaluating the definite integral of a function that has no explicit ____________ or whose antiderivative is not easy to obtain.

antiderivative
Or
derivative

The need often arises for evaluating the definite integral of a function that has no explicit antiderivative or whose antiderivative is not easy to obtain.

• An indefinite integral may _________ in the sense that the limit defining it may not exist.

An improper integral may diverge in the sense that the limit defining it may not exist. In this case, there are more sophisticated definitions of the limit which can produce a convergent value for the improper integral. These are called summability methods.

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127

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