Linear Algebra

358 Chapter Five. SimilarityIISimilarityWe’ve defined two matrices H and Ĥ to be matrix equivalent if there arenonsingular P and Q such that Ĥ = PHQ. We were motivated by this diagramshowing both H and Ĥ representing a map, h but with respect to different pairsof bases, B, D and ˆB, ˆD.V wrt Bid⏐↓V wrt ˆBh−−−−→Hh−−−−→ĤW wrt Did⏐↓W wrt ˆDWe now consider the special case where the codomain equals the domainand in particular we add the requirement that the codomain’s basis equals thedomain’s basis, so we are considering representations with respect to B, B andD, D.tV wrt B −−−−→ V wrt BS⏐⏐id↓id↓V wrt Dt−−−−→TV wrt DIn matrix terms, Rep D,D (t) = Rep B,D (id) Rep B,B (t) ( Rep B,D (id) ) −1.II.1Definition and Examples1.1 Definition The matrices T and S are similar if there is a nonsingular P suchthat T = PSP −1 .Since nonsingular matrices are square, T and S must be square and of the samesize. Exercise 12 checks that similarity is an equivalence relation.1.2 Example Calculation with these two,( )2 1P =1 1gives that S is similar to this matrix.( )0 −1T =1 1( )2 −3S =1 −11.3 Example The only matrix similar to the zero matrix is itself: PZP −1 = PZ = Z.The identity matrix has the same property: PIP −1 = PP −1 = I.