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 λ1/λ2 +i λ1/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 B MatrixSB (e) (f) (a6) (b6) (g) (h) (c6) (d6) Δ(det): tr: Δ(det): tr: (tr/2)2 -Δ (tr/2)2 -Δ anti-tr: anti-tr/2b anti-tr: anti-tr/2b √[(tr/2)2 -Δ] +i √[(tr/2)2 -Δ] +i λ1 +i λ1 +i λ2 +i λ2 +i λ1/λ2 +i λ1/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 (1) & (3) b=c ; d/a = u/y (a1) (b1) (u1r) + i (u1i) (v1) (u1ar) + i (u1ai) (c1) (d1) (w1) (y1r) +i (y1ar) +i(y1ai) Find Similarity matrix satisfying condition (1) & (3) b=c ; v=w (a2) (b2) (u2) (v2r) +   i   (v2i) (v2ar) + i (v2ai) (c2) (d2) (w2r) +   i   (w2i) (y2) (w2ar) + i (w2ai) Find Similarity matrix satisfying STAS=Ka AS=(ST)-1Ka ,   b=c (a5) (b5) (u5) (v5) (c5) (d5) (w5) (y5) Matrix L=SAS-1 Matrix La=S-1AS (L11) (L12) (La11) (La12) (L21) (L22) (La21) (La22) Δ(det): tr: Δ(det): tr: (tr/2)2 -Δ (tr/2)2 -Δ anti-tr: anti-tr/2L12: anti-tr: anti-tr/2La12: √[(tr/2)2 -Δ] +i √[(tr/2)2 -Δ] +i Find Similarity matrix SA satisfying condition S-1AS=La or AS=SLa ....(4) (a3) (b3) (u3) (v3) (c3) (d3) (w3)+i() (y3)+i() Δ(det): tr: (tr/2)2 -Δ anti-tr: anti-tr/2b √[(tr/2)2 -Δ] +i λ1 +i λ2 +i Find Similarity matrix satisfying condition SAS-1=L or SA=LS ....(5) (a4) (b4) (u4) (v4) (c4) (d4) (w4) (y4)