Refer a friend and earn $50 when they subscribe to an annual planRefer Now

Get the answer to your homework problem.

Try Numerade Free for 30 Days

Like

Report

Another estimate can be made for an eigenvalue when an approximate eigenvector is available. Observe that if $A \mathbf{x}=\lambda \mathbf{x}$ , then $\mathbf{x}^{T} A \mathbf{x}=\mathbf{x}^{T}(\lambda \mathbf{x})=\lambda\left(\mathbf{x}^{T} \mathbf{x}\right),$ and the Rayleigh quotient$$R(\mathbf{x})=\frac{\mathbf{X}^{T} A \mathbf{x}}{\mathbf{x}^{T} \mathbf{x}}$$equals $\lambda$ . If $\mathbf{x}$ is close to an eigenvector for $\lambda,$ then this quotient is close to $\lambda .$ When $A$ is a symmetric matrix $\left(A^{T}=A\right)$ , the Rayleigh quotient $R\left(\mathbf{x}_{k}\right)=\left(\mathbf{x}_{k}^{T} A \mathbf{x}_{k}\right) /\left(\mathbf{x}_{k}^{T} \mathbf{x}_{k}\right)$ will have roughly twice as many digits of accuracy as the scaling factor $\mu_{k}$ in the power method. Verify this increased accuracy in Exercises 11 and 12 by computing $\mu_{k}$ and $R\left(\mathbf{x}_{k}\right)$ for $k=1, \ldots, 4$$$A=\left[\begin{array}{ll}{5} & {2} \\ {2} & {2}\end{array}\right], \mathbf{x}_{0}=\left[\begin{array}{l}{1} \\ {0}\end{array}\right]$$

The actual eigenvalue is 6. The bottom two columns of the table show that $R\left(\mathbf{x}_{\mathbf{k}}\right)$ estimates the eigenualue more accurately than $\mu_{k}$ .

Calculus 3

Chapter 5

Eigenvalues and Eigenvectors

Section 8

Iterative Estimates for Eigenvalues

Vectors

Johns Hopkins University

Missouri State University

Oregon State University

University of Michigan - Ann Arbor

Lectures

02:56

In mathematics, a vector (…

06:36

01:34

Another estimate can be ma…

02:20

Let $A$ be as in Exercise …

02:12

In Exercises 19 and $20,$ …

01:43

In Exercises $1-4,$ the ma…

02:17

A widely used method for e…

07:07

Determine whether or not $…

17:43

Find an eigenvector associ…

35:11

Deal with the eigenvalue/e…

20:00

Let $A=\left[\begin{array}…

okay, in this question, we basically want to compare two different ways off calculating Eigen balance. So the traditional way in the power method is that we set you okay? Should be that the entry. You okay with the biggest A k a r a X k. Yeah. I x k was biggest the guest. You guess. Absolutely. Valley, In this case, though, we want to compare this between using this and realize questions which is defined as X transpose a X over x transpose X. So this new case is calculated over here, and this realized politician, which I called here is basically this thing right here. So ex transfers well supplied by a X and X transpose X right there. So the important thing to recognize is that in both cases, we are using the same estimated by Becca. It's just we're simply comparing the idea of values, so yeah, so I paid a plant that's good right here, basically comparing the u. K and R K. And this is the results. And you can see that the realize proficient he has high person

View More Answers From This Book

Find Another Textbook

In mathematics, a vector (from the Latin word "vehere" meaning &qu…

In mathematics, a vector (from the Latin "mover") is a geometric o…

Another estimate can be made for an eigenvalue when an approximate eigenvect…

Let $A$ be as in Exercise 9 . Use the inverse power method with $\mathbf{x}_…

In Exercises 19 and $20,$ find $(a)$ the largest eigenvalue and $(b)$ the ei…

In Exercises $1-4,$ the matrix $A$ is followed by a sequence $\left\{\mathbf…

A widely used method for estimating eigenvalues of a general matrix $A$ is t…

Determine whether or not $\mathbf{x}$ is an eigenvector of $A .$ If it is, d…

Find an eigenvector associated with the given eigenvalue of $A .$$$\begi…

Deal with the eigenvalue/eigenvector problem for $n \times n$ real skew-symm…

Let $A=\left[\begin{array}{rrr}{4} & {-1} & {-1} \\ {-1} & {4} &…

04:00

Let $f, g,$ and $h$ be linearly independent functions defined for all real n…

01:55

Let $X$ be the design matrix in Example 2 corresponding to a least-squares f…

01:46

Suppose $\mu$ is an eigenvalue of the $B$ in Exercise $15,$ and that $\mathb…

03:21

Let $\left\{y_{k}\right\}$ be the sequence produced by sampling the continuo…

03:01

Let $V$ be a vector space with a basis $\mathcal{B}=\left\{\mathbf{b}_{1}, \…

Suppose the $x$ -coordinates of the data $\left(x_{1}, y_{1}\right), \ldots,…

07:42

Diagonalize the matrices in Exercises $7-20,$ if possible. The eigenvalues f…

00:50

Determine which sets of vectors are orthonormal. If a set is only orthogonal…

03:40

In Excrcises $1-6,$ the given set is a basis for a subspace $W .$ Use the Gr…

03:41

Show that $\left\{\mathbf{u}_{1}, \mathbf{u}_{2}\right\}$ or $\left\{\mathbf…

92% of Numerade students report better grades.

Try Numerade Free for 30 Days. You can cancel at any time.

Annual

0.00/mo 0.00/mo

Billed annually at 0.00/yr after free trial

Monthly

0.00/mo

Billed monthly at 0.00/mo after free trial

Earn better grades with our study tools:

Textbooks

Video lessons matched directly to the problems in your textbooks.

Ask a Question

Can't find a question? Ask our 30,000+ educators for help.

Courses

Watch full-length courses, covering key principles and concepts.

AI Tutor

Receive weekly guidance from the world’s first A.I. Tutor, Ace.

30 day free trial, then pay 0.00/month

30 day free trial, then pay 0.00/year

You can cancel anytime

OR PAY WITH

Your subscription has started!

The number 2 is also the smallest & first prime number (since every other even number is divisible by two).

If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

Receive weekly guidance from the world's first A.I. Tutor, Ace.

Mount Everest weighs an estimated 357 trillion pounds

Snapshot a problem with the Numerade app, and we'll give you the video solution.

A cheetah can run up to 76 miles per hour, and can go from 0 to 60 miles per hour in less than three seconds.

Back in a jiffy? You'd better be fast! A "jiffy" is an actual length of time, equal to about 1/100th of a second.