= Stretches. In analogy with the polar decomposition w = x + iy pe of complex...

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= Stretches. In analogy with the polar decomposition w = x + iy pe of complex numbers, any matrix A can be decomposed A = RS into a stretch S and a rotation R; the building blocks of all linear operators. Show that two eigenvectors V₁, Vj of S, with distinct eigenvalues X₁ X₁, must be orthogonal ( Vj = 0); and, therefore, that the matrix of eigenvectors is an orthogonal transformation (V*V = I). Bonus: Interpret the resulting decompositions S = VWV and W = VSV in terms of rotations. that Calculate the eigenvalues and eigenvectors of M₂ =(1-1). Do they look familiar? Show 1/1 1 - (-) (679) (7) = 2 0 and M₂ eM₂ - = VWVt ¹-² (1-4) (8) (17) i i 0 VeWoyt = Stretches. In analogy with the polar decomposition w = x + iy pe of complex numbers, any matrix A can be decomposed A = RS into a stretch S and a rotation R; the building blocks of all linear operators. Show that two eigenvectors V₁, Vj of S, with distinct eigenvalues X₁ X₁, must be orthogonal ( Vj = 0); and, therefore, that the matrix of eigenvectors is an orthogonal transformation (V*V = I). Bonus: Interpret the resulting decompositions S = VWV and W = VSV in terms of rotations. that Calculate the eigenvalues and eigenvectors of M₂ =(1-1). Do they look familiar? Show 1/1 1 - (-) (679) (7) = 2 0 and M₂ eM₂ - = VWVt ¹-² (1-4) (8) (17) i i 0 VeWoyt

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ANSWER Proof Lets assume that there exist two eigenvectors of S denoted as u and v with distinct eigenvalues and Therefore we have Su u and Sv v We ca... View the full answer