Question

Problem 4. Let A, B e Rmxn. We say that A is equivalent to B if there exist an invertible m x m n x n matrix Q such that PAQ = B. matrix P and an invertible (a) Prove that the relation "A is equivalent to B" is reflexive, symmetric, and transitive; i.e., prove that: (i) for all A E Rmx", A is equivalent to A; (ii) for all A, B e Rmxn, if A is equivalent to B then B is equivalent to A;* (iii) for all A, B, C E R"x", if A is equivalent to B and B is equivalent to C, then A is equivalent to C (Ъ) Prove that for all A, B E RMXN, A and B are equivalent if and only if rank(A) = rank(B).t

Answer 1

_mxn A ER ImA In=A Since Im and In ane non-singular matrices, we must have A n equivalent to A. Therafore given Teladiun u Teflexive i) Let for A,B ER" mxn equiralent to B. invertible x in matdx p and an A Then thereexists 5nvertie hxn matnx a sueh that PAQ= B PA- BQ non sing wiar, P and g" are also nonsingukae repectiraly Since p amd a ane Omd p and Q have order m and n Thore fore given relatn e Dequivaleni to A. Hemce Hte Symmetc B i equiialot i) tet -for A,B, CER"^ and equivalent A in S equival:e nt fo B, ev exist an fnver tible mam matix P sueh teat Since A is PAR= B nxn matix invantible and an an invertible mxm matrix thare eist equivdent to Since X B y= c nan matrix Y sneh that and an fnvert'ble x (pAQ)Y-C XBY C P) A (aY) = C Simce x Ond p ane nn Singusan, Xp ùalso nonsingnlar omd order Since &amd Y ane nonsingusea, aY i also nunsirgnlar and urder ar n There fore A in equialent to C e hawe Hence he 4ransihive girem velalfon is Consequenty the given Yelathien i on eauivalenee velatiar.

b) Let Mxn A eauivalent to B. wlere A,B ER Then PA R, fr singulan maticen P omd SoYne Υιση res peetively order man Tharefare ramk( A) rank (B) Conversely et fur A/BER Yank(A) = Tank(B) = say) Then eaeh of A and B equivalent ta tue mxn ana equialent ana Rorefura A and B