Hamilton wrote three short papers on linear operators on quaternions, showing how the inverse of such an operator can be expressed as a polynomial in the operator, and proving a result for linear operators on the space of quaternions which is a special case of the general theorem which today is known as the `Cayley-Hamilton Theorem', which states that every square matrix and every linear operator on a finite-dimensional vector space satisfies its characteristic equation.
The paper
On a New and General Method of Inverting a Linear and Quaternion Function of a Quaternionwas communicated on June 9th, 1862, and appeared in the Proceedings of the Royal Irish Academy, volume 8 (1864), pp. 182-183.
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The paper
On the Existence of a Symbolic and Biquadratic Equation, which is satisfied by the Symbol of Linear Operation in Quaternionswas communicated on June 23rd, 1862, and appeared in the Proceedings of the Royal Irish Academy, volume 8 (1864), pp. 190-191.
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The paper
On the Existence of a Symbolic and Biquadratic Equation, which is satisfied by the Symbol of Linear or Distributive Operation on a Quaternionwas published in the Philosophical Magazine, volume 24 (4th series) (1862), pp. 127-128.
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D.R. Wilkins