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Linear algebra is one of the more well-behaving parts of math. Yet, I wonder if there occur any pathologies in finite-dimensional linear algebra?

By a pathology I mean a phenomenon whose properties are considered atypically bad or counterintuitive, see Wikipedia.

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Isn't that a bit a matter f taste? Some might consider things like the zero-dimensional space of $0\times 0$ matrices pathological ... –  Hagen von Eitzen Nov 15 '12 at 20:58
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Community wiki? –  Nikola Milinković Nov 16 '12 at 0:40
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5 Answers 5

Non-diagonalizable matrices could be considered pathological.

Matrices that aren't full rank could be considered pathological.

Nonzero symmetric nilpotent matrices could be considered pathological.

However, whether one finds such things to be pathological depends in how much one relies on the properties these violate.

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I thought symmetric nilpotent matrices must be zero. –  chaohuang Nov 15 '12 at 22:59
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It only happens in positive characteristic: e.g. in characteristic 2, you have $$\left( \begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix} \right)$$ –  Hurkyl Nov 15 '12 at 23:16
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It's perhaps not a pathology, but when eigenvalues have (geometric multiplicity) $\neq$ (algebraic multiplicity), you need to be a little more careful than otherwise.

Another (non)pathology might be given by the Hilbert matrix, which is ill-conditioned and hence notoriously sensitive to rounding errors in numerical computations.

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The highest-voted answer to mathoverflow.net/questions/16829/… is the matrix $\pmatrix{0&1\cr0&0\cr}$. –  Gerry Myerson Nov 15 '12 at 23:04
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Gaussian elimination with partial pivoting is in pathological cases explosively unstable, yet in practice is utterly stable.

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Consider the (not especially pathological) map $T:{\bf R}^2\to{\bf R}^2$ given by $$T\pmatrix{x\cr y\cr}=\pmatrix{2&1\cr1&1\cr}\pmatrix{x\cr y\cr}$$ If you view this as a map from the unit square to itself (that is, take a point in the unit square, multiply it by that $2\times2$ matrix, and then replace each entry in the answer with its fractional part to get back into the unit square), then the map is chaotic. It's known as "the cat map".

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Nonsymmetric positive definite matrices are kind of pathological.

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