Linear Algebra · Advanced · 9 min
The directions that stay
Most vectors turn under a matrix. Eigenvectors stay on their own line, possibly reversing or collapsing to zero.
WHAT HAPPENS IF…Try a 90° rotation. Can a nonzero real vector stay on its line?
Preparing your experiment…
Real eigenvalues solve λ² − tr(A)λ + det(A) = 0. A negative discriminant means no real eigenvectors. A repeated eigenvalue has one eigenline unless A is a scalar multiple of the identity, when every nonzero vector is an eigenvector.
A detail that matters
A zero eigenvalue is allowed, but the zero vector itself is never an eigenvector.
Keep the connection going
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