75032 Matrisemetoder 15. august 1995: Fasit

1a

N (A) er utspent av [\begin{matrix}1 \\ 2 \\ 3 \end{matrix}] . a_{3} = −\frac{1}{3}a_{1} − \frac{2}{3}a_{2}.

1b

Aa_{1} = −\frac{11}{3}a_{1} − \frac{10}{3}a_{2}, Aa_{2} = \frac{13}{3}a_{1} + \frac{11}{3}a_{2}, B = \frac{1}{3} [\begin{matrix}−11 & 13 \\ −10 & 11 \end{matrix}] .

1c

λ ∈ {0, i, −i}.

2a

Q_{1} = \frac{1}{3\sqrt{2}} [\begin{matrix}0 & 1 & ∗ & ∗ \\ 3 & −2 & ∗ & ∗ \\ 0 & 3 & ∗ & ∗ \\ 3 & 2 & ∗ & ∗ \end{matrix}] , Σ = [\begin{matrix}4 & 0 \\ 0 & 3 \\ 0 & 0 \\ 0 & 0 \end{matrix}] , Q_{2} = \frac{1}{\sqrt{2}} [\begin{matrix}1 & 1 \\ 1 & −1 \end{matrix}] .

2b

x = \frac{1}{36} [\begin{matrix}17 \\ 1 \end{matrix}] .

3a

u_{k} = A^{k}u_{0} + (I − A)^{−1}(I − A^{k})a.

4

1.

5

e^{A} = (cos α)I + \frac{sin α}{α}A.