Similarity Transformations

(2x2 real matrix)

 Matrix A Matrix S (a) (b) (u) (v) (c) (d) (w) (y) Δ(det): tr: Δ(det): tr: (tr/2)2 -Δ (tr/2)2 -Δ anti-tr: anti-tr/2b anti-tr: anti-tr/2v √[(tr/2)2 -Δ] +i √[(tr/2)2 -Δ] +i λ1 +i λ1 +i λ2 +i λ2 +i (y/x)1: +i (y/x)1: +i (y/x)2: +i (y/x)2: +i (y/x): +i (y/x): +i Matrix K=SAST Matrix Ka=STAS (K11) (K12) (Ka11) (Ka12) (K21) (K22) (Ka21) (Ka22) Δ(det): tr: Δ(det): tr: (tr/2)2 -Δ (tr/2)2 -Δ anti-tr: anti-tr/2k12 anti-tr: anti-tr/2ka12 √[(tr/2)2 -Δ] +i √[(tr/2)2 -Δ] +i λ1 +i λ1 +i λ2 +i λ2 +i (y/x)1: +i (y/x)1: +i (y/x)2: +i (y/x)2: +i (y/x): +i (y/x): +i Find Similarity matrix satisfying condition (2) & (4) (a1) (b1) (u1) (v1) (c1) (d1) (w1) (y1) Find Similarity matrix satisfying condition (1) & (3) b=c ; d/a = u/y √(1-ad/b2)=± i (a1a) (b1a) (u1a) (v1ar) - i(v1ai) (c1a) (d1a) (w1ar) + i(w1ai) (y1a) Matrix L=SAS-1 Matrix La=S-1AS (L11) (L12) (La11) (La12) (L21) (L22) (La21) (La22) (a2) (b2) +i (u2) +i (v2) + i(u2) + i(v2) (c2) (d2) +i (w2) (y2) + i(w2)