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Chapter 7: Q18E (page 336)

Consider the matrix A=|abba|where aand bare arbitrary constants. Find all eigenvalues of A.

Short Answer

Expert verified

Eigenvalue of A=位1,2=a卤b

Step by step solution

01

Eigenvalues

In linear algebra, an eigenvector or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by , is the factor by which the eigenvector is scaled.

02

Step 2: Find all eigenvalues of A

We can clearly see that,

detA-位l=0a-位bb-a-位=0a-位-b2=0位1,2=2a卤4a2-4a2-b22=a卤b

Hence, 位1,2=a卤b.

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Most popular questions from this chapter

Consider ann脳n matrix such that the sum of the entries in each row is . Show that the vector

e鈫�=11..1

InRn is an eigenvector of A. What is the corresponding eigenvalue?

Consider the linear space Vof alln脳nmatrices for which all the vectorse鈬赌1,.....,e鈬赌nare eigenvectors. Describe the spaceV(the matrices inV"have a name"), and determine the dimension ofV.

For each of the matrices in Exercises 1 through 13, find all real eigenvalues, with their algebraic multiplicities. Show your work. Do not use technology.

[010001100]

Find a 2脳2matrix A such that [31]and [12] are eigenvectors of A , with eigenvalues 5 and 10 , respectively.

Three holy men (let鈥檚 call them Anselm, Benjamin, and Caspar) put little stock in material things; their only earthly possession is a small purse with a bit of gold dust. Each day they get together for the following bizarre bonding ritual: Each of them takes his purse and gives his gold away to the two others, in equal parts. For example, if Anselm has 4 ounces one day, he will give 2 ounces each to Benjamin and Caspar.

(a) If Anselm starts out with 6 ounces, Benjamin with 1 ounce, and Caspar with 2 ounces, find formulas for the amounts a(t), b(t), and c(t) each will have after tdistributions.

Hint: The vector [111],[1-10]and[10-1], and will be useful.

(b) Who will have the most gold after one year, that is, after 365 distributions?
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