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Chapter 7: Q7-54E (page 384)

TRUE OR FALSE

54. All symmetric2脳2matrices are diagonalizable (over鈩�)

Short Answer

Expert verified

The given statement is true.

Step by step solution

01

Definition of Diagonalizable matrix

If the square matrix of order n has n distinct eigenvalues, then that matrix is diagonalizable.

02

Determine the given statement is true or false

Let A=abbd.

To find the eigenvalues of A, we solve:

det(A-位I)=0a-位bbd-位=0(a-位)(d-位)-b2=0位2-(a+d)位+ad-b2=0位1,2=a+d卤a2+2ad+d2-4ad-b22=a+d卤(a-d)2+b22

Ifa鈮�dor b鈮�0, we have two distinct real eigenvalues of a 2x2 matrix, so A is diagonalizable.

However, ifa=dand b=0, thenA=d00ais already diagonal, so it's diagonalizable.

Therefore, the given statement is True.

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Most popular questions from this chapter

consider an eigenvalue 位0of ann脳nmatrix A. we are told that the algebraic multiplicity of exceeds 1.Show thatf'(位0)=0(i.e.., the derivative of the characteristic polynomial of A vanishes are位0).

28 : Consider the isolated Swiss town of Andelfingen, inhabited by 1,200 families. Each family takes a weekly shopping trip to the only grocery store in town, run by Mr. and Mrs. Wipf, until the day when a new, fancier (and cheaper) chain store, Migros, opens its doors. It is not expected that everybody will immediately run to the new store, but we do anticipate that 20% of those shopping at Wipf鈥檚 each week switch to Migros the following week. Some people who do switch miss the personal service (and the gossip) and switch back: We expect that 10% of those shopping at Migros each week go to Wipf鈥檚 the following week. The state of this town (as far as grocery shopping is concerned) can be represented by the vector

x炉(t)=[wtm(t]]

where w(t) and m(t) are the numbers of families shopping at Wipf鈥檚 and at Migros, respectively, t weeks after Migros opens. Suppose w(0) = 1,200 and m(0) = 0.

a. Find a 2 脳 2 matrix A such that role="math" localid="1659586084144" x炉(t++1)=Ax鈫�(t). Verify that A is a positive transition matrix. See Exercise 25.

b. How many families will shop at each store after t weeks? Give closed formulas. c. The Wipfs expect that they must close down when they have less than 250 customers a week. When does that happen?

For a given eigenvalue, find a basis of the associated eigenspace .use the geometric multiplicities of the eigenvalues to determine whether a matrix is diagonalizable.

For each of the matrices A in Exercise1 through 20 , find all (real)eigenvalues. Then find a basis of each eigenspaces, and diagonalize A, if you can. Do not use technology.

[7809]

Find a basis of the linear space V of all 2脳2matrices Afor which both[11]and[12]are eigenvectors, and thus determine the dimension of.

Prove the part of Theorem 7.2.8 that concerns the trace: If an n 脳 n matrix A has n eigenvalues 位1, . . . , 位n, listed with their algebraic multiplicities, then tr A = 位1+路 路 路+位n.
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