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Math 324: Assignment 8
Instructions. On this assignment, you may use MathCAD for things like evaluating deter- minants, or rref, multiplying matrices and finding inverses. Do not use its built-in commands for finding eigenvalues or eigenvectors. This assignment is a group assignment. The entire class must agree upon and turn-in one assignment. It is worth 30 homework points and due Monday, November 8.
- (10 pts) Consider the linear operator T (x, y, z) = (7x − 4 y + 10z, 4 x − 3 y + 8z, − 2 x + y − 2 z).
(a) Represent T as a matrix with respect to the standard basis on IR^3.
(b) Find the characteristic polynomial of T , then find the eigenvalues and eigenvectors for the matrix you found in (a).
(c) Find a basis β on IR^3 for which D = [T ]β is a diagonal matrix.
(d) Evaluate T (2, − 3 , 1) using the formula given above.
(e) Find the coordinate vector [v]β where v = (2, − 3 , 1) and β is the basis in (c).
(f) Evaluate [T v]β = [T ]β [v]β by performing the matrix multiplication on the right hand side. Convert the coordinate [T v]β back to standard form. Does this agree with your answer in (d)?
(g) Find a formula for An^ where A is the matrix representation of T you found in (a). Check your formula with n = 5.
(h) Use (g) to find a formula for T n(x, y, z).
(i) Evaluate p(A) where p is the characteristic polynomial of T and A is the matrix represen- tation of T found in (a).
(j) Find the characteristic polynomial of D the diagonal matrix you found in (c). Is it the same as the characteristic polynomial of A?
Answer. Done on MathCAD.
- (3 pts) Show that if A and B are similar matrices, then A and B have the same characteristic polynomial.
Proof. Because A and B are similar, there is a matrix C such that A = C−^1 BC. Therefore,
fA(t) = |A − tI| = |C−^1 BC − tI| = |C−^1 BC − tI(C−^1 C)| = |C−^1 BC − C−^1 (tI)C| = |C−^1 (B − tI)C|
= |C−^1 | |B − tI| |C| =
|C|
|B − tI||C|
= |B − tI| = fB (t)
That is, A and B have the same characteristic polynomial.
Note, also, that question 2 implies that similar matrices have the same eigenvalues since the characteristic polynomials have the same zeros.
- (4 pts) (a) Find bases for the eigenspaces of A =
(b) Is A diagonalizable? Justify your answer.
Answer. (a) The eigenvalues of A are λ 1 = 1 and λ 2 = 2. Note, Eλ 1 = {v ∈ IR^4 : (A−1I) = 0}. A basis for Eλ 1 is a basis for the homogeneous system of equations (A − I)v = 0 which is {(1, 0 , 0 , 0)}. Similarly, for λ 2 = 2, Eλ 2 = {v ∈ IR^4 : (A − 2I) = 0. You should check that a basis for Eλ 2 = {(1, 1 , 0 , 0), (1, 0 , 1 , 0)}.
(b) A is not diagonalizable because dim(Eλ 1 ) + dim(Eλ 2 ) = 1 + 2 = 3 < 4.
- (6 pts) (a) The Fibonacci sequence is defined by the difference equation
an = an− 1 + an− 2
where a 1 = 1, and a 2 = 1. Find a formula for an.
(b) Determine the exact value lim n→∞
an an− 1
. (This limit is called the golden ratio.)
Answer. MathCAD did not do this question very cleanly. Here is another approach. Let bn = an− 1 , then an = an− 1 + bn− 1 and bn = an− 1. Therefore, in matrix form, [ an bn
]
[
] [
an− 1 bn− 1
]
hence,
[ a 3 b 3
]
[
] [
] [
a 4 b 4
]
[
] [
a 3 b 3
]
[
] 2 [
]
And in general (see notes from class on solving difference equations),
[ an an− 1
]
[
]n− 2 [ 1 1
]
Now let A =
[
]
. Then the eigevalues of A are λ 1 =
and λ 2 =
. One can
check that
[
λ 1 1
]
and
[
λ 2 1
]
are eigenvectors corresponding to these eigenvalues. Therefore,
using diaglonalization techniques,
An−^2 = B
[
λn 1 −^20 0 λn 2 −^2
]
B−^1 where B =
[
λ 1 λ 2 1 1
]
(b) Describe the eigenvalues of A and justify your answer.
(c) Is the matrix A =
diagonalizable? Why?
Answer. (a) The chracteristic polynomial is
fA(t) = |A − tI| = (a 11 − t)(a 22 − t) · · · (ann − t)
(b) The eigenvalues of A are λk = akk for k = 1, 2 ,... , n, because these are the zeros of the characteristic polynomial in (a).
(c) Yes, because it is a 4 by 4 matrix with 4 distinct eigenvalues 1, 2 , 3 , 4. (It is interesting to compare this with the matrix in question 3 which is the same except for the values on the diagonal. The matrix in question 3 is not diagonalizable, but the matrix in this question is.)
