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Chapter 7: Q7-7E (page 383)

TRUE OR FALSE

7. If the standard vectorse鈫�1,e鈫�2,鈥�,e鈫�nare eigenvectors of ann脳nmatrix A, then A must be diagonal.

Short Answer

Expert verified

True, A is diagonal.

Step by step solution

01

Define eigenvalue

Eigenvalues are a set of specialized scales associated with a system of linear equations. The corresponding eigenvalue is often denoted by位.

Eigenvalues are the unique set of scalar values which are related to a set of linear equations that are most likely seen in matrix equations. On solving a characteristic equation, we get the roots whose another name for the eigenvectors. It is a non-zero vector that, after applying linear transformations, can only be altered by its scalar factor.

02

Explanation for A is diagonal

Consider,eI,i=1,....,n,are eigenvectors of A then it applies,

Aei=位iei, where i=1,鈥�,n.

A=AI

A=位1e1鈥�位nen

So, A is diagonal.

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Most popular questions from this chapter

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

find an eigenbasis for the given matrice and diagonalize:

A=17[6-2-3-23-6-3-6-2]

Representing the reflection about a plane E.

Arguing geometrically, find all eigen vectors and eigen values of the linear transformations. In each case, find an eigen basis if you can, and thus determine whether the given transformation is diagonalizable.

The linear transformation withT(v鈫�)=v鈫�, and T(w鈫�)=v鈫�+w鈫�for the vectorsv鈫�andw鈫�inR2 sketched below.

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?

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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