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Chapter 7: Q26E (page 345)

Show that if a 6 脳 6 matrix A has a negative determinant, then A has at least one positive eigenvalue. Hint: Sketch the graph of the characteristic polynomial

Short Answer

Expert verified

Matrix A has one positive eigen value.

Step by step solution

01

Finding limits of  fA

Consider an n x n matrix and a scalar .

Then is the eigenvalue of A if det(A-位濒)=0.

This is called characteristic equation of matrix A.

We know that fA0detA-0l6=detA鈮�0

We also know that the coefficient of 位6is -16=1,

So im位鈫�鈭�fA位=鈭�

02

Finding continuity of  fA

Here we consider that 位鈫�鈭�.

Since fAis a continuous function,

Then,

鈭�位0,鈭�fA位=0

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Most popular questions from this chapter

23: Suppose matrix A is similar to B. What is the relationship between the characteristic polynomials of A and B? What does your answer tell you about the eigenvalues of A and B?

Consider a 4 脳 4 matrixA=|BC0D|where B, C, and D are 2 脳 2 matrices. What is the relationship among the eigenvalues of A, B, C, and D?

(a) Give an example of a 3 脳 3 matrix A with as many nonzero entries as possible such that both span(e鈫�1) and span(e鈫�1,e鈫�2) are A-invariant subspaces of . See Exercise 65.

(b) Consider the linear space Vof all 3 脳 3 matrices A such that both span (e鈫�1) and span (e鈫�1,e鈫�2) are A-invariant subspaces of R3 . Describe the space V (the matrices in V 鈥渉ave a name鈥�), and determine the dimension of V.

Arguing geometrically, find all eigenvectors and eigenvalues of the linear transformations in Exercises 15 through 22. In each case, find an eigenbasis if you can, and thus determine whether the given transformation is diagonalizable.

Rotation through an angle of 180掳inR2.

Two interacting populations of coyotes and roadrunners can be modeled by the recursive equations

h(t + 1) = 4h(t)-2f(t)

f(t + 1) = h(t) + f(t).

For each of the initial populations given in parts (a) through (c), find closed formulas for h(t) and f(t).

(a)h(0)=f(0)=100(b)h(0)200,f(0)=100(c)h(0)600,f(0)=500
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