316 MATHEMATICSMAGAZINE Proof Referto FIGURE3. We findthat there are twelve pairsof triangles,each pair of which has a commonside or a commonangle (or two congruentangles one from each triangleof the pair).Applying P1 andP2 on these triangleswe have
AM Thus, if we let IA = a, IC = c, IM = m, IN = n, we get
a-m which simplifiesto
1 Hencea butterflyinscribed in a quadrilateralsatisfies the samerelation (1) as a butterfly inscribedin a circle.Equivalently, the conclusionof the theoremindicates that the ratio of the ratios, (AM/IM)/(CN/IN),is the same as the ratioIA/IC, or thatthe harmonic meanof IC andIM equalsthe harmonicmean of IA andIN. In eithercase, if IC = IA, we haveIM = IN therebythe analogof the usualbutterfly theorem for quadrilaterals.
Acknowledgment. The authorwould like to thank the referees for their helpful suggestions, and Joseph Kung for preparationof the article.
REFERENCE
1. Leon Bankoff, The metamorphosisof the butterflyproblem, this MAGAZINE60 (1987), 195-210.
Row RankEquals Column Rank WILLIAM P. WARDLAW U. S. Naval Academy Annapolis, MD 21402-5000 [email protected]
Dedicatedto GeorgeMackiw, a goodfriend and an excellentmathematical expositor
The purpose of this note is to present a short (perhaps shortest?) proof that the row rank of a matrix is equal to its column rank. The proof is elementary and accessi- ble to students in a beginning linear algebra course. It requires only the definition of
Mathematical Association of America is collaborating with JSTOR to digitize, preserve, and extend access to Mathematics Magazine ® www.jstor.org VOL. 78, NO. 4, OCTOBER 2005 317 matrixmultiplication and the fact that a minimalspanning set is a basis. It differsin approachfrom proofs given in textbooksas well as from some interestingproofs in MAAjournals [1, 2]. And,unlike the latter,this proofis validover any field of scalars. But first,recall thatif the m x n matrixA = BC is a productof the m x r matrix B andthe r x n matrixC, thenit follows fromthe definitionof matrixmultiplication thatthe ith row of A is a linearcombination of the r rows of C with coefficientsfrom the ith row of B, andthe jth columnof A is a linearcombination of the r columnsof B with coefficientsfrom the jth columnof C. (If you have troubleunderstanding this or the nextparagraph, you shouldconstruct several examples of smallmatrix products, say, a 3 x 2 times a 2 x 3 matrix,etc., with smallinteger as well as symbolicentries.) On the otherhand, if any collectionof r row vectorsc1, spansthe row space of A, an r x n matrixC can be formedby takingthese vectorsas its rows.Then the ith row of A is a linearcombination of the rows of C, say bil1- birCr. This means A = BC, where B = (bij) is the m x r matrix whose ith row, bir), is formed from the coefficients giving the ith row of A as a linearcombination of the r rows of C. Similarly,if any r columnvectors span the columnspace of A, and B is the m x r matrixformed by these columns,then the r x n matrixC formedfrom the appropriate coefficientssatisfies A = BC. Now the four sentenceproof.
THEOREM.If A is an m x n matrix, then the row rank of A is equal to the column rank of A.
Proof If A = 0, thenthe row andcolumn rank of A areboth 0; otherwise,let r be the smallestpositive integer such thatthere is an m x r matrixB andan r x n matrix C satisfyingA = BC. Thus the r rows of C form a minimalspanning set of the row space of A and the r columnsof B form a minimalspanning set of the columnspace of A. Hence,row andcolumn ranks are both r. E Severalother properties of the rankof a matrixover a field arealso very easy to ob- tain.The factorizationsA = ImA= AInshow that r < m andr < n, whichproves that the rankof A is less thanor equalto the numberof rows andthe numberof columns of A. Since A = BC implies AT = CT BT, the transpose clearly has the same rank as the original matrix. Since A = BC and D - EF implies AD = B(CD) = (AE)F, the rankof AD mustbe less thanor equalto the rankof A andto the rankof D. These conceptssuggest the followingdefinition [5, p. 123]: DEFINITION.Let R be a commutativering with identityand let A be an m x n matrix over R. Then the spanning rank of A is 0 if A = 0 and otherwise is the smallest positive integer r such that there is an m x r matrix B and an r x n matrix C satisfying A = BC.
This definitionis one way of extendingthe notionof rankto matricesover commu- tativerings. Even if the ringhas no identity,it can be embeddedin a ringwith identity so thatthe definitioncan be used.Care must be takenin consideringrank over commu- tative rings, because several different extensions of the definitions over a field can give different results over rings, even though they all give the standardconcept of rank over a field. Nonetheless, if the above definition is used, matrices over rings automatically have row rank equal to column rank, have rank less than or equal to the number of rows and the number of columns, the rank of the transpose is equal to the rank of the matrix, and the rank of a product is less than or equal to the rank of either factor. Another application of the spanning rank, first used by the author in a problem [3] and later a Note [5] in the MAGAZINE, is the proof that a matrix over a commutative 318 MATHEMATICSMAGAZINE ring with spanningrank r satisfiesa polynomialequation of degreeat most r + 1. For if A = BC is an n x n matrix of spanning rank r, then D = CB is an r x r matrix with characteristic polynomial fD (x) = det(xI - D) of degree r and fD(D) = 0 fol- lows fromthe Cayley-HamiltonTheorem. (The authorhas shown [4] thatthe Cayley- HamiltonTheorem holds for matricesover commutativerings.) Thus thereis a poly- nomial m(x) of smallest positive degree such that m(D) = 0. Then p(x) = xm(x) is a polynomial of degree 1 such that p(A) = Am(A) = BCm(BC) = Bm(CB)C = Bm(D)C = 0.
REFERENCES
1. H. Liebeck, A proof of the equalityof column and row rankof a matrix,Amer Math.Monthly 73 (1966), 1114. 2. G. Mackiw, A note on the equality of column and row rankof a matrix,this MAGAZINE68 (1995), 285-286. 3. W. Wardlaw,problem 1179, this MAGAZINE56 (1983), 326, and solution 1179, this MAGAZINE57 (1984), 303. 4. - A transferdevice for matrixtheorems, this MAGAZINE59 (1986), 30-33. 5. - Minimumand characteristicpolynomials of low-rankmatrices, this MAGAZINE68 (1995), 122-127.
A Modern Approachto a Medieval Problem AWANI KUMA R, Director Lucknow Zoological Garden Lucknow 226001 INDIA [email protected]
The following problem from Lilavati [1], a mathematicaltreatise written by Bhaskaracharya,a 12th-centuryIndian mathematicianand astronomer,deserves a modemapproach:
A snake'shole is at the foot of pillar,nine cubitshigh, anda peacockis perched on its summit.Seeing a snakeat the distanceof thricethe pillargliding towards his hole, he pouncesobliquely upon him. Say quicklyat how manycubits from the snake'shole they meet,both proceeding an equaldistance.
Since both proceedan equal distance,it is reasonableto assumethat their speeds are equal.Readers are invitedto solve this problembefore proceeding. Assumingthat the peacockflies alongthe hypotenuseof a right-angledtriangle and knowsthe PythagoreanTheorem, it will grabthe snakeat a distanceof 12 cubitsfrom the pillar.Practically, however, such a thing does not happen.Why shoulda peacock know-a priori-to fly along the hypotenuseof a right-angledtriangle having a base of 12 cubits?A morepeacock-like behavior would be to keepan eagle eye on the snake andchange its directionat everyinstant, always aiming toward the snake. This type of pursuit problem has a history of over five hundred years. However, this particularproblem is a bit different from most. Instead of the prey running away from the predator, here prey and the predator are moving toward each other. Even so, the results are startling. Although the reader may have seen similar problems, I offer a general analysis. We assume that the snake and the peacock move at different, but constant, speeds: the snake in a straight line toward its hole and the peacock along a curve, changing its direction at every instant so as to be flying directly toward the snake.