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For a matrix to have eigenvectors, it must represent a linear transformation of a finite-dimensional vector space into itself and hence must be a square matrix. Note also from the lead that a defective matrix "is therefore not diagonalizable." This is a result—not the definition. The inference
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Does a defective matrix truly need to be square? In other words, non-diagonalizable matrices can be found which are not square (all of these, in fact), and also not diagonalizable, which seems to be the word sense of "defective". Since failing diagonalization is a trait not limited to square
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does not hold in the other direction, that is, a matrix which is not diagonalizable is not necessarily a defective matrix. For more information about eigenvectors, see most any linear algebra textbook. For more information about defective matrices, see Golub and Van Loan.—
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matrices, it seems artificial to limit these matrices to being square. Are there different conventions on this, or is there some other terminology which helps clarify the status of these non-square matrices with respect to the definition of defective matrix?
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A matrix is singular iff one of its eigenvalues is zero. Whether a matrix is defective has to do with the multiplicities of the eigenvalues, and not with the location of the eigenvalues. This explains why the properties are independent. --
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I didn't revert your example entirely. I just changed it to something a bit more simple. I agree that the singular one wasn't so good. This article actually needs a bit of help. There is no mention of the connection to the
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Hi
Plastikspork. I see you reverted my change to the simple example. Could you explain your reasoning? You said "It was already a simple example". My objection to the original example was that it was
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Perhaps I'm overlooking something simple, but isn't a matrix singular iff its determinate is zero? The same appears from the examples to be true of a defective matrix.
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Not. All four combinations (invertible + diagonalizable, invertible + defective, singular + diagonalizable, singlar + defective) are possible. --
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Sorry, I didn't look very carefully at what you'd done - too hasty with my indignation. --
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It's dangerous to generalize from one example :) Here are some others:
377:{\displaystyle {\begin{bmatrix}0&0\\0&1\end{bmatrix}}}
322:{\displaystyle {\begin{bmatrix}1&1\\0&1\end{bmatrix}}}
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I think this page should be merged with its antonym,
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480:. Thanks for your help.
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