/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 10 Let \(A\) be an \(n \times n\) m... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 10

Let \(A\) be an \(n \times n\) matrix and let \(B=A-\alpha I\) for some scalar \(\alpha .\) How do the eigenvalues of \(A\) and \(B\) compare? Explain.

Short Answer

Expert verified
The eigenvalues of matrix B are related to the eigenvalues of matrix A by \(\lambda_B = \lambda_A - \alpha\), where \(\lambda_A\) is an eigenvalue of A, and \(\lambda_B\) is the corresponding eigenvalue of B. This means that the eigenvalues of B are shifted by a scalar 伪 from the eigenvalues of A.

Step by step solution

01

Recall the definition of an eigenvalue and eigenvector

First, let's recall what an eigenvalue and eigenvector are. If A is an n x n matrix, a scalar 位 is called an eigenvalue of A, and a nonzero vector v is called an eigenvector of A, if Av = 位v.
02

Write the eigenvalue equation for A and B

Let 位_A and 位_B be eigenvalues of A and B, respectively, and let v_A and v_B be the corresponding eigenvectors. Then, we have the following eigenvalue equations: \(A v_A = \lambda_A v_A\) and \(B v_B = \lambda_B v_B\)
03

Express B eigenvalue equation in terms of A

We know that B = A - 伪I, so, we can substitute this expression into the eigenvalue equation for B: \((A - \alpha I)v_B = \lambda_B v_B\)
04

Find the relationship between 位_A and 位_B

Now, we want to find a relationship between 位_A and 位_B. From step 3, we know that: \(A v_B = (\lambda_B + \alpha)v_B\) Comparing this with the eigenvalue equation for A (step 2), we realize that they have the same structure, with v_A and v_B being eigenvectors corresponding to the eigenvalues 位_A and 位_B, respectively. Thus, we can conclude that: \(\lambda_B = \lambda_A - \alpha\)
05

Interpret the results

This result tells us the relationship between the eigenvalues of A and B: for any eigenvalue of A, the corresponding eigenvalue of B is obtained by subtracting 伪 from that eigenvalue. In other words, the eigenvalues of B are shifted by 伪 compared to those of A.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Let \(U\) be a unitary matrix. Prove that (a) \(U\) is normal. (b) \(\|U \mathbf{x}\|=\|\mathbf{x}\|\) for all \(\mathbf{x} \in \mathbb{C}^{n}\) (c) if \(\lambda\) is an eigenvalue of \(U,\) then \(|\lambda|=1\)

Let \\[ A=\left(\begin{array}{rr} 1 & -\frac{1}{2} \\ -\frac{1}{2} & 1 \end{array}\right) \quad \text { and } \quad B=\left(\begin{array}{rr} 1 & -1 \\ 0 & 1 \end{array}\right) \\] (a) Show that \(A\) is positive definite and that \(\mathbf{x}^{T} A \mathbf{x}=\mathbf{x}^{T} B \mathbf{x}\) for all \(\mathbf{x} \in \mathbb{R}^{2}\) (b) Show that \(B\) is positive definite, but \(B^{2}\) is not positive definite.

Let \\[ A=\left(\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right) \\] Write \(A\) as a sum \(\lambda_{1} \mathbf{u}_{1} \mathbf{u}_{1}^{T}+\lambda_{2} \mathbf{u}_{2} \mathbf{u}_{2}^{T},\) where \(\lambda_{1}\) and \(\lambda_{2}\) are eigenvalues and \(\mathbf{u}_{1}\) and \(\mathbf{u}_{2}\) are orthonormal eigenvectors.

Let \(A\) be a \(3 \times 3\) symmetric positive definite matrix and suppose that \(\operatorname{det}\left(A_{1}\right)=3, \operatorname{det}\left(A_{2}\right)=6,\) and \(\operatorname{det}\left(A_{3}\right)=8 .\) What would the pivot elements be in the reduction of \(A\) to triangular form, assuming that only row operation III is used in the reduction process?

For each of the following matrices, compute the determinants of all the leading principal submatrices and use them to determine whether the matrix is positive definite: (a) \(\left(\begin{array}{rr}2 & -1 \\ -1 & 2\end{array}\right)\) (b) \(\left(\begin{array}{ll}3 & 4 \\ 4 & 2\end{array}\right)\) (c) \(\left(\begin{array}{rrr}6 & 4 & -2 \\ 4 & 5 & 3 \\ -2 & 3 & 6\end{array}\right)\) (d) \(\left(\begin{array}{rrr}4 & 2 & 1 \\ 2 & 3 & -2 \\ 1 & -2 & 5\end{array}\right)\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.