Numerical Methods & Linear Algebra

Math 2890-003

(1 point) Let \[\{y_0,y_1,y_2, \ldots \}= \{-3,4,2,-5,5,2,0,-5,-4,-3,3,0,5, \ldots \}.\] Use the filter \[z_k = 4y_{k+3}-3y_{k+2}-4y_{k+1}+3y_{k}\] to find the first $6$ terms of the signal $\{z_0, z_1, z_2, \ldots \}$. Show your work.

(1 point) Let \[A=\left(\begin{array}{rrrr} -154 & 112 & 58 & -36 \\ -183 & 133 & 68 & -42 \\ -198 & 144 & 77 & -48 \\ -226 & 164 & 86 & -53 \end{array}\right).\] Compute $A^{9}.$ Show and explain your work. Hint: It may help to know that $AP = PD$ where \[P=\left(\begin{array}{rrrr} 1 & 4 & 2 & 2 \\ 1 & 5 & 4 & 1 \\ 2 & 4 & -3 & 9 \\ 2 & 5 & -1 & 9 \end{array}\right),\quad D = \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)\quad\text{and}\quad P^{-1} = \left(\begin{array}{rrrr} -43 & 32 & 18 & -12 \\ 14 & -10 & -5 & 3 \\ -7 & 5 & 2 & -1 \\ 1 & -1 & -1 & 1 \end{array}\right).\]

(1 point) Let \[A=\left(\begin{array}{rrrr} 0.6 & 0.2 & 0.5 & 0.1 \\ 0.1 & 0.6 & 0.1 & 0.3 \\ 0.1 & 0.1 & 0.3 & 0.1 \\ 0.2 & 0.1 & 0.1 & 0.5 \end{array}\right).\] Find a steady state probability vector for the stochastic matrix $A$. Show and expain your work.

(1 point) Consider \[y_{k+2}-4y_{k+1}-12y_{k}=0.\] Find the general solution of this linear difference equation. Show and explain your work.

(1 point) Consider \[y_{k+2}+11y_{k+1}+28y_{k}=200.\] Find the general solution of this linear difference equation. Show and explain your work.

(1 point) Consider \[y_{k+2}-25y_{k}=-36(-4)^k.\] Find the general solution of this linear difference equation. Show and explain your work.

(1 point) Consider \[y_{k+2}+5y_{k+1}-6y_{k}=28k+4.\] Find the general solution of this linear difference equation. Show and explain your work.