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A non-derogatory matrix $A$ is one, whose minimal polynomial $m(z)$ equals its characteristic polynomial $p(z)$, where we apply the convention $p(z) = det(zI-A)$, while a matrix is derogatory, if they do not coincide. I have certainly never felt particularly offended when dealing with the identity matrix $I$, so seriously I wonder, where that strange name has its origin.

Some background:

I would like to know about it, because I teach some advanced pupils at a german gymnasium school, who will be considering a field of study in the STEM-area, some linear algebra using examples from polynomial geometry. In that context, certain polynomials generate matrices with a double eigenvalue, which are either non-derogatory or diagonalizable, and where the non-derogatory ones are indeed preferable to the diagonalizable ones due to the particular application.

In order to make the rather involved notion more palatable to the audience, it is always good to have a story to tell about it, even if it turns out to be rather dull in the end.

My native language is german and I know of only one place in the german literature, where non-derogatory matrices get a special name. That is in E.Brieskorn "Lineare Algebra und analytische Geometrie II", where they are called "regular", which the author immediately distinguishes from another use of that word as invertible.

On the contrary, the english denomination sems to be well-established when refering to Google and I found it already mentioned in the first chapter of J.H. Wilkinson "The Algebraic Eigenvalue Problem" from 1965 without further comment, so it seems to have been established by then.

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    $\begingroup$ The term seems self-explanatory (=deficient matrix, non-diagonalizable matrix). And the question on historical usage is better to ask in History of Science and Maths SE. $\endgroup$ – Alexandre Eremenko Apr 5 at 19:37
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The original meaning is derived from Latin derogare, "to take away, detract from, diminish", see EtymOnline. Association with offense is from later, the 16th century. For matrices, the term was introduced by Sylvester in the early 1880s, in his matrix based reworking of the Hamilton's theory of quaternions, see The Emergence of the American Mathematical Research Community, 1876-1900, by Parshall and Rowe, p. 136. The "diminishment" probably refers to the degree of the characteristic polynomial being diminished in the minimal polynomial. On the other hand, Sylvester's another suggestion, to rename characteristic into "latent" polynomial, did not catch on.

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  • $\begingroup$ Thank you very much! $\endgroup$ – thomashennecke Apr 6 at 9:45

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