284 CHAPTER 12. SPECTRAL THEORY

36. Suppose the migration matrix for three locations is .1 .1 .3.3 .7 0.6 .2 .7

 .

Find a comparison for the populations in the three locations after a long time.

37. You own a trailer rental company in a large city and you have four locations, one inthe South East, one in the North East, one in the North West, and one in the SouthWest. Denote these locations by SE,NE,NW, and SW respectively. Suppose youobserve that in a typical day, .8 of the trailers starting in SE stay in SE, .1 of thetrailers in NE go to SE, .1 of the trailers in NW end up in SE, .2 of the trailers in SWend up in SE, .1 of the trailers in SE end up in NE,.7 of the trailers in NE end up inNE,.2 of the trailers in NW end up in NE,.1 of the trailers in SW end up in NE, .1of the trailers in SE end up in NW, .1 of the trailers in NE end up in NW, .6 of thetrailers in NW end up in NW, .2 of the trailers in SW end up in NW, 0 of the trailersin SE end up in SW, .1 of the trailers in NE end up in SW, .1 of the trailers in NWend up in SW, .5 of the trailers in SW end up in SW. You begin with 20 trailers ineach location. Approximately how many will you have in each location after a longtime? Will any location ever run out of trailers?

38. Let A be the n×n, n > 1, matrix of the linear transformation which comes from theprojection v7→projw (v). Show that A cannot be invertible. Also show that A has aneigenvalue equal to 1 and that for λ an eigenvalue, |λ | ≤ 1.

39. Let v be a unit vector in Rn and let A = I− 2vvT . Show that A has an eigenvalueequal to −1.

40. Let M be an n× n matrix and suppose x1, · · · ,xn are n eigenvectors which form alinearly independent set. Form the matrix S by making the columns these vectors.Show that S−1 exists and that S−1MS is a diagonal matrix (one having zeros ev-erywhere except on the main diagonal) having the eigenvalues of M on the maindiagonal. When this can be done the matrix is diagonalizable. This is presented inthe text. You should write it down in your own words filling in the details withoutlooking at the text.

41. Show that a matrix M is diagonalizable if and only if it has a basis of eigenvectors.Hint: The first part is done in Problem 40. It only remains to show that if the matrixcan be diagonalized by some matrix S giving D = S−1MS for D a diagonal matrix,then it has a basis of eigenvectors. Try using the columns of the matrix S. Like thelast problem, you should try to do this yourself without consulting the text. Theseproblems are a nice review of the meaning of matrix multiplication.

42. Suppose A is an n×n matrix which is diagonally dominant. This means

|aii|> ∑j ̸=i

∣∣ai j∣∣ .

Show that A−1 must exist.