1-find-eigen-values-and-corresponding-eigen-vector-of-the-matrix-A-cos-sin-sin-cos-2-solve-6y-2-dx-x-y-2x-2-dy-0-

Question Number 91321 by M±th+et+s last updated on 29/Apr/20

1) f i n d e i g e n v a l u e s a n d c o r r e s p o n d i n g

e i g e n v e c t o r o f t h e m a t r i x

A = [\begin{matrix} c o s (θ) & - s i n (θ) \\ s i n (θ) & c o s (θ) \end{matrix}]

2) s o l v e

6 y^{2} d x - x (y + 2 x^{2}) d y = 0

Commented by mathmax by abdo last updated on 30/Apr/20

1) P (x) = d e t (A - x I) = | \begin{matrix} c o s θ - x - s i n θ \\ s i n θ c o s θ - x \end{matrix} |

= {(c o s θ - x)}^{2} + {s i n}^{2} θ

P (x) = 0 \Leftrightarrow {(x - c o s θ)}^{2} + {s i n}^{2} θ = 0 \Leftrightarrow {(x - c o s θ)}^{2} = {(i s i n θ)}^{2} \Rightarrow

x - c o s θ = \bar{+} i s i n θ \Rightarrow x = c o s θ \bar{+} i s i n θ s o t h e v a l u e s a r e

λ_{1} = e^{i θ} a n d λ_{2} = e^{- i θ}

V (λ_{1}) = K e r (A - λ_{1} I) = {u / (A - λ_{1} I) u = 0}

u (\begin{matrix} x \\ y \end{matrix}) (A - λ_{1}) u = 0 \Rightarrow (\begin{matrix} c o s θ - e^{i θ} - s i n θ \\ s i n θ c o s θ - e^{i θ} \end{matrix}) (\begin{matrix} x \\ y \end{matrix}) = 0 \Rightarrow

(\begin{matrix} - i s i n θ - s i n θ \\ s i n θ - i s i n θ \end{matrix}) (\begin{matrix} x \\ y \end{matrix}) = 0 \Rightarrow {\begin{cases} - i s i n θ x - s i n θ y = 0 \\ s i n θ x - i s i n θ y = 0 \end{cases}

\Rightarrow {\begin{cases} i s i n θ x + s i n θ y = 0 \\ s i n θ x - i s i n θ y = 0 \Rightarrow s i n θ x = i s i n θ y \end{cases}

i f s i n θ \neq 0 \Rightarrow x = i y \Rightarrow (x, y) = (i y, y) = y e_{1} w i t h e_{1} (\begin{matrix} i \\ 1 \end{matrix})

V (λ_{2}) = K e r (A - λ_{2} I) = {u / (A - λ_{2} I) u = 0}

(A - λ_{2} I) u = 0 \Rightarrow (\begin{matrix} c o s θ - e^{- i θ} - s i n θ \\ s i n θ c o s θ - e^{- i θ} \end{matrix}) (\begin{matrix} x \\ y \end{matrix}) = 0 \Rightarrow

(\begin{matrix} i s i n θ - s i n θ \\ s i n θ i s i n θ \end{matrix}) (\begin{matrix} x \\ y \end{matrix}) = 0 \Rightarrow {\begin{cases} i s i n θ x - s i n θ y = 0 \\ s i n θ x + i s i n θ y = 0 \end{cases}

\Rightarrow s i n θ x = - i s i n θ y s o i f s i n θ \neq 0 w e g e t x = - i y \Rightarrow

(x, y) = (- i y, y) = {y e}_{2} w i t h e_{2} (\begin{matrix} - i \\ 1 \end{matrix})

W e s e e V (λ_{1}) = Δ_{e_{1}} d i m (V (λ_{1})) = 1

V (λ_{2}) = Δ_{e_{2}} a n d d i m (V (λ_{2})) = 1

A i s d i a g o n a l i s a b l e \dots .

Commented by M±th+et+s last updated on 30/Apr/20

n i c e a n d c o r r e c t w o r k s i r g o d b l e s s y o u

Commented by turbo msup by abdo last updated on 30/Apr/20

y o u a r e w e l c o m e .