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Chapter 7: Q57E (page 384)

TRUE OR FALSE

If v1,v2,.....,vnis an eigen basis for both Aand B, then matrices Aand Bmust commute.

Short Answer

Expert verified

The given statement is true.

Step by step solution

01

Definition of Eigen basis

A basis of the set of all the vectors, in which each vector is an eigenvector, is known as the eigen basis.

02

Determine the given statement is true or false

Let S = [v1,.....,vn ]

Then, for matrix A and B, we have,

AS = [伪1v1....伪nvn ]

BS =[尾1v1....尾nvn ]

which gives us

ABS = [伪1尾1v1....伪n尾nvn]

Since S is invertible, leads to AB=BA.

Therefore, the given statement is true.

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Most popular questions from this chapter

Find a 2脳2matrixsuch that

X鈫�(t)=[2t-6t2t+6t]is a trajectory of the dynamical systemrole="math" localid="1659527385729" x鈫�(t+1)=Ax鈫�(t)

27: a. Based on your answers in Exercises 24 and 25, find closed formulas for the components of the dynamical system

x炉(t+1)=[0.50.250.50.75]x炉(t)

with initial value x0鈫�=e1鈫�. Then do the same for the initial value x0鈫�=e2鈫�. Sketch the two trajectories.

b. Consider the matrix

A=[0.50.250.50.75]

.

Using technology, compute some powers of the matrix A, say, A2, A5, A10, . . . .What do you observe? Diagonalize matrix Ato prove your conjecture. (Do not use Theorem 2.3.11, which we have not proven

yet.)

c. If A=[abcd]

is an arbitrary positive transition matrix, what can you say about the powers Atas t goes to infinity? Your result proves Theorem 2.3.11c for the special case of a positive transition matrix of size 2 脳 2.

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?

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.

[5-421]

For an arbitrary positive integer n, give a 2n脳2n matrix A without real eigenvalues.
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