Algebra II 189-251B Instructor: Dr. E. Goren.

Assignment 11

To be submitted by April 13, 12:00

1. 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} ) .

2. 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.

3. Find a formula for the general term of the sequence

0, 1, 4, 12, 32, 80, …

(a_{n+2} = 4a_{n+1} − 4a_{n}).

4. Find an orthogonal matrix P such that P^{−1}AP is diagonal, where

A = ( \begin{matrix}2 & 1 & 1 & 1 \\ 1 & 2 & 1 & 1 \\ 1 & 1 & 2 & 1 \\ 1 & 1 & 1 & 2 \end{matrix} ) .