Publisher's Synopsis
Excerpt from On the Roots of Matrices
In his memoir on Matrices (phil. Trans. 1858) Prof. Cayley enunciated the theorem: The determinant, having for its matrix a given matrix less the same matrix considered as a single quantity involving the matrix unity, is equal to zero. The equation implied in this theorem is known as Cayley's identical equation. Subsequently (in the Mess. Math. Vol. XIII, p. Mr. A. R. Forsyth gave a proof of this identical equation for matrices of the third order, based upon the solution of a system of linear difference equations.* Forsyth's method is applicable to matrices of any order. Considerable simplicity is gained, however, by the employment of non-scalar equations instead of the scalar equa tions employed by Forsyth.
I have employed this modification of Forsyth's method to prove Sylvester's law of latency and Sylvester's theorem. In addition I have by this method investigated the existence of 'roots of matrices for different indices and in particular the roots of nilpotent matrices.
For valuable suggestions in the working of this paper I am indebted to Dr. Henry Taber.
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