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My take: The eigenvectors are the “axes” of the transformation represented by the matrix. Consider spinning a globe (the universe of vectors): every location
by kalid 12y ago
My take:
The eigenvectors are the “axes” of the transformation represented by the matrix.
Consider spinning a globe (the universe of vectors): every location faces a new direction, except the poles.
An “eigenvector” is an input that doesn’t change direction when it’s run through the matrix (it points “along the axis”). And although the direction doesn’t change, the size might. The eigenvalue is the amount the eigenvector is scaled up or down when going through the matrix.
(Shameless plug, more here: http://betterexplained.com/articles/linear-algebra-guide/ http://betterexplained.com/articles/linear-algebra-guide/)
- JadeNB 12y agoBe careful using this to interpret complex eigenvalues, though! (One could argue that a complex eigenvector points along the same complex axis, but that probably doesn't help much to understand the eigenvalues of a real matrix like ( ( 0 1 ) ( 1 0 ) ).)
- wging 12y agoWho says ((0 1) (1 0)) is a real matrix? It's a complex matrix whose entries all happen to have 0 as the value of their imaginary part :)
- JadeNB 12y agoI didn't say that it isn't complex, only that it is real—and as such can be visualised as a transformation of the real plane, in which context one might still strive to understand its (complex) eigendecomposition.
- bgilroy26 12y agoWhat's a kernel?
- throwaway5752 12y agohttp://en.wikipedia.org/wiki/Kernel_%28linear_algebra%29 http://en.wikipedia.org/wiki/Kernel_%28linear_algebra%29 If you have a linear transformation from one vector space to another, the kernel is the part of the domain that maps to the zero vector in the range. Intuitively, if you have a function that satisfies certain properties, the kernel is the part that it collapses to "zero".