We state and prove the invariance, with respect to matrix power, of both the diagonals and anti-diagonals ratio of a special case 2×2 matrix. The proof methodology is new, contrasting with those deployed previously in establishing anti-diagonals matrix invariants.N/

Fennessey, Eric J.

Larcombe, Peter J.

We state and prove the invariance, with respect to matrix power, of both the diagonals and anti-diagonals ratio of a special case 2×2 matrix. The proof methodology is new, contrasting with those deployed previously in establishing anti-diagonals matrix invariants

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Palestine Journal of MathematicsVol. 7(2)(2018) , 410–413 © Palestine Polytechnic University-PPU 2018A NOTE ON TWO RATIONAL INVARIANTSFOR A PARTICULAR 2 × 2 MATRIXPeter J. Larcombe and Eric J. FennesseyCommunicated by Ayman BadawiMSC 2010 Classifications: Primary 11C20.Keywords and phrases: Anti-diagonals and diagonals ratio invariance.Abstract We state and prove the invariance, with respect to matrix power, of both the diag-onals and anti-diagonals ratio of a special case 2 × 2 matrix. The proof methodology is new,contrasting with those deployed previously in establishing anti-diagonals matrix invariants.1 Introduction1.1 BackgroundLetN = N(A,B,C,D) =(A BC D)(1.1)be a general 2 × 2 matrix, assuming A,B,C,D ̸= 0. A somewhat surprising result holds—namely, that (unless otherwise indeterminate) the ratio of the two anti-diagonal terms in anyexponentiated instance of N is the quantity B/C—which is apparently little known in the liter-ature. In a previous publication [2] the observation was formulated in four different ways. Ourshort note offers a new proof of this occurrence and a similar one which, by necessity, both applyin the case when D = A and so to a slightly less general matrix than N above.1.2 The ResultWe begin by decomposing N(A,B,C,D) as N(A,B,C,D) = F(A,D) + S(B,C), whereF(A,D) =(A 00 D)and S(B,C) =(0 BC 0). (1.2)Then Nn(A,B,C,D) = [F(A,D) + S(B,C)]n =∑nk=0 (nk)Fn−k(A,D)Sk(B,C) only if thematrices F(A,D) and S(B,C) commute. It is elementary to see that this occurs when D = A,which is the imposition mentioned above. Thus, F(A,D) simplifies to F = F(A) = AI2(denoting the 2 × 2 identity matrix by I2), with S(B,C) as in (1.2) andF(A) + S(B,C) = M(A,B,C), (1.3)say, whereM(A,B,C) = N(A,B,C,A) =(A BC A), (1.4)and for which we will prove these invariance results:Theorem 1.1. Unless otherwise indeterminate, the ratio of the two anti-diagonal terms in Mn isthe quantity B/C while the ratio of the two diagonal terms therein is unity, both being invariantwith respect to integer power n ≥ 1.TWO RATIONAL INVARIANTS FOR A 2 × 2 MATRIX 4112 The ProofProof. Consider, from (1.3),Mn(A,B,C) = [F(A) + S(B,C)]n=n∑k=0(nk)Fn−k(A)Sk(B,C)=∑k(even)≥0(nk)Fn−k(A)Sk(B,C) +∑k(odd)≥1(nk)Fn−k(A)Sk(B,C)=∑m≥0(n2m)Fn−2m(A)S2m(B,C)+∑m≥0(n2m+ 1)Fn−(2m+1)(A)S2m+1(B,C); (P.1)the index m takes values according to the parity of n, upper limits for which being a detail notrequired in the proof. Noting that Fp(A) = ApI2 (integer power p ≥ 0) and, for m = 0, 1, 2, . . . ,S2m(B,C) = (BC)mI2 with S2m+1(B,C) = (BC)mS(B,C), we continue asMn(A,B,C) =∑m≥0(n2m)An−2mI2 · (BC)mI2+∑m≥0(n2m+ 1)An−(2m+1)I2 · (BC)mS(B,C)=∑m≥0(n2m)An−2m(BC)mI2+∑m≥0(n2m+ 1)An−(2m+1)(BC)mS(B,C)= Ωe(A,B,C;n) + Ωo(A,B,C;n), (P.2)where, writing ( n2m)An−2m(BC)m = gm(n) = gm(A,B,C;n) and ( n2m+1)An−(2m+1)(BC)m =fm(n) = fm(A,B,C;n),Ωo(A,B,C;n) =∑m≥0fm(n)S(B,C) =(0 B∑m≥0 fm(n)C∑m≥0 fm(n) 0)(P.3)is anti-diagonal, withΩe(A,B,C;n) =∑m≥0gm(n)I2 =( ∑m≥0 gm(n) 00∑m≥0 gm(n))(P.4)diagonal. It is now immediate from (P.2)-(P.4) that the anti-diagonals ratio of Mn is precisely thatof Ωo and is B/C, and that the unity diagonals ratio of Ωe is that of Mn also, both independentof matrix power n.Combinations of A,B,C that induce ratio indeterminacy are not discussed here, although someare immediate by inspection of the first few powers of M which are provided in Appendix A andalso illustrate Theorem 1.1.Remark 2.1. For the purpose of completeness we note that the authors’ attention has been drawnto a 2005 paper by Cisneros-Molina where [1, Lemma 3.2, p. 148] the entries of a fully generalexponentiated 2×2 matrix are listed in closed form and from which Theorem 1.1 is an immediatecorollary (see Appendix B)—this, though, does not diminish our proof here in any way.412 Peter J. Larcombe and Eric J. Fennessey3 SummaryThis note presents proofs of the diagonals and anti-diagonals ratio invariance for a particular 2×2matrix, complementing previous work on the fully general case in which the anti-diagonals ratioinvariance was stated [2, Theorem 1.1, p. 360] and proven, and the diagonals ratio invariancewas merely noted (p. 362 therein). The line of argument used here does not, however, seemapplicable to an examination of the invariance of those anti-diagonals ratios within an arbitrarydimension tri-diagonal matrix addressed in [3], where background information on the topic ofmatrix anti-diagonals ratio invariance may be found along with references to closed form entriesfor exponentiated matrices (see also [2]).AcknowledgementThe authors are grateful to Dr. James Clapperton for undertaking all computations referred to inthe article.Appendix AThe initial powers of the matrix M (1.4) are calculated to be the following:M2 =(A2 +BC 2AB2AC A2 +BC),M3 =(A(A2 + 3BC) B(3A2 +BC)C(3A2 +BC) A(A2 + 3BC)),M4 =(A4 + 6A2BC +B2C2 4AB(A2 +BC)4AC(A2 +BC) A4 + 6A2BC +B2C2),M5 =(A(A4 + 10A2BC + 5B2C2) B(5A4 + 10A2BC +B2C2)C(5A4 + 10A2BC +B2C2) A(A4 + 10A2BC + 5B2C2)),M6 =((A2 +BC)(A4 + 14A2BC +B2C2) 2AB(A2 + 3BC)(3A2 +BC)2AC(A2 + 3BC)(3A2 +BC) (A2 +BC)(A4 + 14A2BC +B2C2)),... (A.1)With reference to the proof of Section 2, we here check the entries above in a couple of se-lected instances. We have, for example, that∑m≥0 gm(5) =∑m≥0 (52m)A5−2m(BC)m =(50)A5(BC)0 + (52)A3(BC)1 + (54)A1(BC)2 = 1 · A5 · 1 + 10 · A3 · (BC) + 5 · A · (BC)2 =A(A4 + 10A2BC + 5B2C2), seen as the expected diagonal entries of M5. The common ele-ment to the anti-diagonal entries of M6 is∑m≥0 fm(6) =∑m≥0 (62m+1)A6−(2m+1)(BC)m =(61)A5(BC)0+(63)A3(BC)1+(65)A1(BC)2 = 6A5+20A3BC+6AB2C2 = 2A(3A4+10A2BC+3B2C2) = 2A(A2 + 3BC)(3A2 + BC); the interested reader is invited to verify further onessimilarly.Appendix BHere we verify Theorem 1.1 solely by appeal to the results of [1] (note that these formulas arenot derived as such in the article, but we have checked them for matrix powers up to and beyond500 using computer software).Writing the exponentiated general matrix N asNn(A,B,C,D) =(A BC D)n=(αn βnγn δn), (B.1)TWO RATIONAL INVARIANTS FOR A 2 × 2 MATRIX 413closed forms for αn = αn(A,B,C,D), . . . , δn = δn(A,B,C,D) are listed by Cisneros-Molinaasαn = An +⌊n/2⌋∑s=1n−2s∑m=0(n− s−ms)(m+ s− 1m)An−2s−mBsCsDm,βn =⌊(n−1)/2⌋∑s=0n−2s−1∑m=0(n− s−m− 1s)(m+ sm)An−2s−m−1Bs+1CsDm,γn =⌊(n−1)/2⌋∑s=0n−2s−1∑m=0(n− s−m− 1s)(m+ sm)An−2s−m−1BsCs+1Dm,δn = Dn +⌊n/2⌋∑s=1n−2s∑m=0(n− s−m− 1s− 1)(m+ sm)An−2s−mBsCsDm. (B.2)By inspection, the anti-diagonals ratio βn/γn = B/C (regardless of the constraint D = A, andequally so when it applies). In the case when D = A then the diagonals ratio αn/δn is unitysince now αn = δn because the inner sums of αn, δn in (B.2), namelyS1(n; s) =n−2s∑m=0(n− s−m− 1s− 1)(m+ sm)(B.3)andS2(n; s) =n−2s∑m=0(n− s−ms)(m+ s− 1m), (B.4)are identical; this is readily seen by a reversal in the order of summation in S1(n; s) according toS1(n; s) =∑n−2sm=0 (n−s−m−1s−1 )(m+sm ) =∑n−2sm′=0 (n−s−(n−2s−m′)−1s−1 )((n−2s−m′)+sn−2s−m′ ), which reducesto∑n−2sm′=0 (m′+s−1s−1 )(n−s−m′n−2s−m′) =∑n−2sm′=0 (m′+s−1m′ )(n−s−m′s ) = S2(n; s).References[1] J. L. Cisneros-Molina, An invariant of 2 by 2 matrices, Elec. J. Lin. Alg. 13, 146–152 (2005).[2] P. J. Larcombe, A note on the invariance of the general 2 × 2 matrix anti-diagonals ratio with increasingmatrix power: four proofs, Fib. Quart. 53, 360–364 (2015).[3] P. J. Larcombe and E. J. Fennessey, A new tri-diagonal matrix invariance property, Palest. J. Math. 7,9–13 (2018).Author informationPeter J. Larcombe, Department of Computing and Mathematics, College of Engineering and Technology, Uni-versity of Derby, Kedleston Road, Derby DE22 1GB, U.K.E-mail: p.j.larcombe@derby.ac.ukEric J. Fennessey, BAE Systems Surface Ships Ltd., HM Naval Base, Portsmouth PO1 3NJ, U.K.E-mail: eric.fennessey@baesystems.comReceived: June 23, 2017.Accepted: October 5, 2017.

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A note on two rational invariants for a particular 2 x 2 matrix.

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A note on two rational invariants for a particular 2 x 2 matrix.

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