On Differentiating a Matrix

H. W. Turnbull

doi:10.1017/S0013091500007434

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The theorem is well known. So also is the theorem that if concerning a determinant Λ and its reciprocal expressed by means of cofactors Aij of aij. Not quite so well known is the Cayley Hamilton theorem that a matrix X =[xij] satisfies its own characteristic equation

Unlike as these three results are, they nevertheless can be looked upon as particular phases of a general theorem concerning a matrix differential operator acting upon a function of a matrix X or its transposed.

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