5. Condition NumberIn lecture and the course reader, we examined the sensitivity of solutions to linear system Az = y (fornonsingular/invertible square matrix A) to perturbations in our measurements y. Specifically, we showed that ifwe model measurement noise Ay as a linear perturbation on y, resulting in a linear perturbation A on 7-i.e.,A(z+Ax) =ý + Ay- we can bound the magnitude of the solution perturbations Az as||AT||2||||₂maxShow that||||| 2||||₂≤K(A)!where K(A) =is the condition number of A, or the ratio of A's maximum and minimum singularvalues. In this problem, we will establish a similar bound for perturbations on A.(a) Consider the linear system A = y above, where A = R¹x is invertible (i.e., square and nonsingular).Let AA € Rnxn denote a linear perturbation on matrix A generating a corresponding linear perturbationAz in solution 7, i.e.,(Α + ΔΑ)(x + Δt) = y.||AT||2|| +AT||₂(11)||ΔΑ||2|||A||2≤K(A).(12)(13)||Δ.||2||Δ.7|2||||₂ || +A||₂(b) Note that Equations (11) and (13) above bound two slightly different quantities:andrespectively. In general, we wish to establish these bounds because we want to characterize the size ofA under different sizes of perturbation. Which of these two bounds better serves this purpose? HINT:Consider different relative values of and A. What happens to the bounds when Ax» ?

Consider a linear system .Given that, if we model measurement noise as a linear perturbation on ,…

In lecture and the course reader, we examined the sensitivity of solutions to linear system A y (for nonsingular/invertible square matrix A) to perturbations in our measurements 7. Specifically, we showed that if we model measurement noise Ay as a linear perturbation on y, resulting in a linear perturbation A on i.e., A(z+Ax) = ý + A- we can bound the magnitude of the solution perturbations Az as ≤K(A). max |||||2 ||||2 Show that where K(A) = is the condition number of A, or the ratio of A's maximum and minimum singular values. In this problem, we will establish a similar bound for perturbations on A. ||||2 . ||||2 (a) Consider the linear system A = above, where A E Rnxn is invertible (i.e., square and nonsingular). Let AA € Rnxn denote a linear perturbation on matrix A generating a corresponding linear perturbation Az in solution , i.e., (Α + ΔΑ)(* + Δ.π) = g. ||AT||2 || +AT||₂ (11) ||Δ.Α||2 ||A||2 ≤K(A). (12) (13) and (b) Note that Equations (11) and (13) above bound two slightly different quantities: ||A||2 ||Δ.||2 ||||₂ || +A||₂ respectively. In general, we wish to establish these bounds because we want to characterize the size of A under different sizes of perturbation. Which of these two bounds better serves this purpose? HINT: Consider different relative values of and A. What happens to the bounds when ▲x » ?

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.6: Applications And The Perron-frobenius Theorem

Problem 70EQ