Eigenvectors and eigenvalues | Chapter 14, Essence of linear algebra
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Further reading 3
Why shift a matrix before looking for an eigenvector? →A zero determinant signals collapse into a lower dimension. Applying that test to the shifted matrix explains how a nonzero vector can lie in its null space.How do rotations change in four dimensions? →Compare a fixed axis in three dimensions with planes of rotation in four. The familiar axis description does not carry over unchanged.Which outputs disappear when a transformation collapses space? →Some target vectors fall outside the image after a collapse. In the eigenvalue example it is A minus lambda times the identity that becomes singular; this does not make every system unsolvable.
Caption notes
An eigenvector stays on its own line. Finding when A−λI collapses a dimension gives the eigenvalue condition; an eigenbasis simplifies repeated transformations.
Original passages · 4
some special vectors do remain on their own spanmake this determinant zero, meaning the tweaked transformation squishes space into a lower dimensionwhat you have found is the axis of rotationchange to an eigenbasis, compute the 100th power in that system, then convert back