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Chapter 7: Q7.5-46E (page 375)

For which values of the real constant 鈥榓鈥� are the matricesin Exercises 45 through 50 diagonalizable over C?

46.[0-aa0]

Short Answer

Expert verified

If a鈮�0two distinct eigenvalues

If a=0the matrix 0 so A is been diagonala鈭�鈩�

Step by step solution

01

Define Matrix and Diagonalizable

A matrix is a rectangular array of numbers or symbols which are generally arranged in rows and columns.

Inlinear algebra, asquare matrix {\displaystyle A}is called diagonalizable or non-defective if it issimilar to adiagonal matrix.

02

Solving the Equation

The matrix diagonalizable over C

A=0-aa0det(A-位I)=00-aa0-位1001=0-位-aa-位=0-位-aa-位=0位2+a2=0位1,2=卤ia

Therefore,

Ifa鈮�0, two distinct eigenvalues

If a=0, the matrix 0 so A is been diagonal a鈭�鈩�.

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Most popular questions from this chapter

Find all 4x4matrices for whiche鈬赌2is an Eigen-vector.

if A is a 2脳2matrix with t r A = 5and det A = - 14what are the eigenvalues of A?

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.

24: Find all eigenvalues of the positive transition matrix

A=[0.50.250.50.75]

See Definitions 2.1.4 and 2.3.10.

suppose a certain4脳4 matrix A has two distinct real Eigenvalues. what could the algebraic multiplicities of These eigenvalues be? Give an example for each possible Case and sketch the characteristic polynomial.
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