Algebra II MATH 251 Instructor: Dr. E. Goren.

Assignment 10

To be submitted by April 4, 12:00

1. Let v_{1}, …, v_{n} be a basis for V and let T : V ⟶ V be a linear transformation. For every 1 ≤ i ≤ n we define a polynomial P_{i} as follows: Let r be the first integer so that v_{i}, T v_{i}, …, T^{r}v_{i} are linearly dependent. Then for suitable scalars b_{0}, …, b_{r} we have b_{r}T^{r}v_{i} + …, b_{1}T v_{i} + b_{0} = 0. We may assume without loss of generality that b_{r} = 1. Let P_{i}(t) = t^{r} + b_{r−1}t^{r−1} + ⋯ + b_{1}t + b_{0} (the b_{j} depend on i but we do not make this dependence explicit in the notation). Let P = lcm{P_{1}, …, P_{n}}. Prove that P is the minimal polynomial of T.

2. Prove that the matrix of complex numbers

A = (\begin{matrix}1 & 1 & 5 & 0 & 0 \\ 7 & 1 & 3 & 0 & 0 \\ 2 & 1 & −1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \end{matrix})

is diagonalizable. Note: you don’t need to diagonalize it to prove it !!

3. Determine the possibilities for the Jordan canonical form of a matrix A with characteristic polynomial Δ_{A}(t) = (t − 1)^{6}(t − 2)^{4}(t − 4)^{5} and minimal polynomial m_{A}(t) = (t − 1)^{3}(t − 2)^{2}(t − 4).

4. Find the Jordan canonical form of the matrix

A = (\begin{matrix}5 & 9 & −2 & 0 & 0 \\ −1 & −1 & 1 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 5 & 9 \\ 0 & 0 & 0 & −1 & −1 \end{matrix}) .

5. It is known that a differentiable function f : ℝ^{2} ⟶ ℝ has a • maximum at a point P if ∂f/∂x = ∂f/∂y = 0 at P and the 2 × 2 symmetric matrix

− (\begin{matrix}\frac{∂^{2}f}{∂x^{2}} & \frac{∂^{2}f}{∂x∂y} \\ \frac{∂^{2}f}{∂x∂y} & \frac{∂^{2}f}{∂y^{2}} \end{matrix} )

is positive definite;

• minimum at a point P if ∂f/∂x = ∂f/∂y = 0 at P and the 2 × 2 symmetric matrix

(\begin{matrix}\frac{∂^{2}f}{∂x^{2}} & \frac{∂^{2}f}{∂x∂y} \\ \frac{∂^{2}f}{∂x∂y} & \frac{∂^{2}f}{∂y^{2}} \end{matrix} )

is positive definite; • saddle point at P if the 2 × 2 symmetric matrix

(\begin{matrix}\frac{∂^{2}f}{∂x^{2}} & \frac{∂^{2}f}{∂x∂y} \\ \frac{∂^{2}f}{∂x∂y} & \frac{∂^{2}f}{∂y^{2}} \end{matrix} )

has one negative eigenvalue and one positive eigenvalue.

If P is either a maximum, minimum or saddle point, we call it a simple critical point. Determine the nature of the simple critical point of the following functions at the origin (0, 0)

f(x, y) = 2x^{2} + 6xy + y^{2}, f(x, y) = x sin(x) − cos(y) − xy.

(You may view the graphs and rotate them in Maple using plot3d(2*xˆ 2+6*x*y+yˆ 2, x= -4..4, y = -4..4); plot3d(x*sin(x) - cos(y) -x*y, x= -4..4, y = -4..4, numpoints=3000); ).

Remark. This criterion can be generalized to functions f : ℝ^{n} ⟶ ℝ. If all the first partials vanish at a point P and the matrix of mixed derivatives is positive definite (resp. negative definite) at P, then the function has a minimum (resp. maximum) at P.

6. Find a formula for the general term of the sequence 0, 1, 4, 12, 32, 80, …

(a_{n+2} = 4a_{n+1} − 4a_{n}).

Note: The point here would be that the matrix A that comes out is not diagonalizable. Still you should find some matrix D = MAM^{−1} simple enough so that the trick A^{N} = M^{−1}D^{N}M is still useful.