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XXV.—On Bipartite Functions
Published online by Cambridge University Press: 06 July 2012
Extract
If a row of n elements be taken, and closely following this array, but separated by a bar from it, we write n rows of n elements each; and closely following either outside column of this square array, but separated by a bar from it, we write n columns of n elements each; and closely following an outside row of this second square array, but separated by a bar from it, we write n rows of n elements each; and so on, passing from the rows or columns of one array to the columns or rows of the next, and ending not with a square array, but, as we began, with a single line of elements, we have the matrix representation of a bipartite function.
- Type
- Research Article
- Information
- Earth and Environmental Science Transactions of The Royal Society of Edinburgh , Volume 32 , Issue 3 , February 1886 , pp. 461 - 482
- Copyright
- Copyright © Royal Society of Edinburgh 1886
References
page 481 note * After the theory of this new class of functions had heen worked out under a temporary designation of my own, I got the Philosophical Transactions for 1858, in consequence of a communication on another matter from Professor Tait, in order to read Professor Cayley's Memoir on Matrices; and there found, immediately following the said memoir, another, “On the Automorphic Linear Transformation of a Bipartite Quadric Function.” This quadric function I saw at the first glance was a member of the class I had been dealing with—viz., that of the third degree. This led me to discard the name I had been employing, and to adopt bipartite instead. Professor Cayley gives the above extension of the theorem regarding the invariance of the discriminant of a quadric, but without proof, and not as if looking at it from that point of view. I think, however, I am correct in saying that this is the only point in which my paper has heen anticipated. Professor Cayley's notation for the bipartite we have used above is
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20190116055958330-0991:S0080456800025229:S0080456800025229_eqnU1.gif?pub-status=live)
which does not, I think, bear on the face of it the exact nature of the two-sidedness of a bipartite of the third degree; that is to say, it does not imply, as
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20190116055958330-0991:S0080456800025229:S0080456800025229_eqnU2.gif?pub-status=live)
does, that the function is equal to
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20190116055958330-0991:S0080456800025229:S0080456800025229_eqnU3.gif?pub-status=live)
It may be of interest, as another evidence of the usefulness of bipartites, to remark here that the “Memoir on Matrices” came opportunely for another reason. The new instrument I had got hold of seemed as if specially devised for dealing with matrices, and I immediately succeeded in proving Cayley's great theorem that, if M be a matrix, the equation—
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20190116055958330-0991:S0080456800025229:S0080456800025229_eqnU4.gif?pub-status=live)
is satisfied by
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20190116055958330-0991:S0080456800025229:S0080456800025229_eqnU5.gif?pub-status=live)
This proof, with its accessories, has been communicated to the Mathematical Society of London.