Respuesta :
Matrix of t = [tex]\left[\begin{array}{ccc}0&-1\\-1&0\\\end{array}\right][/tex] and eigenvalues are 1 and -1 Eigenvectors [tex](\left[\begin{array}{ccc}x\\y\\\end{array}\right] )=k(\left[\begin{array}{ccc}1\\-1\\\end{array}\right] )[/tex].
As given in the question,
Linear transformation which rotates all vectors r2 counterclockwise α= π/2.
Required matrix = [tex]\left[\begin{array}{ccc}cos\alpha &-sin\alpha \\-sin\alpha &cos\alpha \end{array}\right][/tex]
= [tex]\left[\begin{array}{ccc}cos\pi /2 &-sin\pi/2 \\-sin\pi /2 &cos\pi /2\end{array}\right][/tex]
= [tex]\left[\begin{array}{ccc}0&-1\\-1&0\\\end{array}\right][/tex]
Eigen values λ
[tex]\left[\begin{array}{ccc}0&-1\\-1&0\\\end{array}\right][/tex] -λ[tex]\left[\begin{array}{ccc}1&0\\0&1\\\end{array}\right][/tex] =0
⇒ λ²-1 =0
⇒λ =±1
Eigenvectors
AX=λX
⇒(A-λI) X=0
⇒[tex](\left[\begin{array}{ccc}0&-1\\-1&0\\\end{array}\right]-1\left[\begin{array}{ccc}1&0\\0&1\\\end{array}\right])(\left[\begin{array}{ccc}x\\y\\\end{array}\right] )=(\left[\begin{array}{ccc}0\\0\\\end{array}\right] )[/tex]
⇒-x-y=0
⇒x=k and y =-k
Eigenvectors [tex](\left[\begin{array}{ccc}x\\y\\\end{array}\right] )=k(\left[\begin{array}{ccc}1\\-1\\\end{array}\right] )[/tex]
Therefore, matrix of t = [tex]\left[\begin{array}{ccc}0&-1\\-1&0\\\end{array}\right][/tex] and eigenvalues are 1 and -1 Eigenvectors [tex](\left[\begin{array}{ccc}x\\y\\\end{array}\right] )=k(\left[\begin{array}{ccc}1\\-1\\\end{array}\right] )[/tex].
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