11.6. EXERCISES 305

7. Find a formula for An where

A =

2 − 1

212 −1

4 0 1 −452 − 1

2 1 −23 − 1

212 −2

Note that the rows sum to 1 in this matrix also. Hint: This matrix is not similar to adiagonal matrix but you can find the Jordan form and consider this in order to obtaina formula for this product. The eigenvalues are 1,−1, 1

2 ,12 .

8. Find limn→∞ An if it exists for the matrix

A =

12 − 1

2 − 12 0

− 12

12 − 1

2 012

12

32 0

32

32

32 1

The eigenvalues are 1

2 ,1,1,1.

9. Give an example of a matrix A which has eigenvalues which are either equal to 1,−1,or have absolute value strictly less than 1 but which has the property that limn→∞ An

does not exist.

10. If A is an n× n matrix such that all the eigenvalues have absolute value less than 1,show limn→∞ An = 0.

11. Find an example of a 3×3 matrix A such that limn→∞ An does not exist but limr→∞ A5r

does exist.

12. If A is a Markov matrix and B is similar to A, does it follow that B is also a Markovmatrix?

13. In Theorem 11.1.3 suppose everything is unchanged except that you assume either∑ j ai j ≤ 1 or ∑i ai j ≤ 1. Would the same conclusion be valid? What if you don’tinsist that each ai j ≥ 0? Would the conclusion hold in this case?

14. Let V be an n dimensional vector space and let x ∈V and x ̸= 0. Consider

βx ≡ x, Ax, · · · ,Am−1x

whereAmx ∈ span

(x,Ax, · · · ,Am−1x

)and m is the smallest such that the above inclusion in the span takes place. Showthat

{x,Ax, · · · ,Am−1x

}must be linearly independent. Next suppose {v1, · · · ,vn}

is a basis for V . Consider βvias just discussed, having length mi. Thus Amivi is a

linearly combination of vi,Avi, · · · ,Am−1vi for m as small as possible. Let pvi (λ ) bethe monic polynomial which expresses this linear combination. Thus pvi (A)vi = 0and the degree of pvi (λ ) is as small as possible for this to take place. Show thatthe minimum polynomial for A must be the monic polynomial which is the leastcommon multiple of these polynomials pvi (λ ).