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Chapter 7: Q45E (page 375)

For which values of the real constant a are the matrices diagonalizable over C?

[1a11]

Short Answer

Expert verified

Fora鈮�0, we have two distinct eigenvalues. For the value a=0, the matrix

A=i[1011]s not diagonalizable.

Step by step solution

01

Define Diagonalizable

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

02

Step 2:Solving[1a11]

det(A鈭�位I)=0|1鈭�位1a1鈭�位|=0(1鈭�位)2鈭�a=0位1,2=2卤4鈭�4(1鈭�a)2=1卤a

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Most popular questions from this chapter

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

24: Find all eigenvalues of the positive transition matrix

A=[0.50.250.50.75]

See Definitions 2.1.4 and 2.3.10.

Find all2脳2matrices for whiche鈬赌1is an eigenvector.

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 Exercises 1 through 20, find all (real) eigenvalues. Then find a basis of each eigenspace, and diagonalize A, if you can. Do not use technology

6.(2345)

If v鈬赌is any nonzero vector in R2 , what is the dimension of the space Vof all 2脳2matrices for which v鈬赌is an eigenvector?
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