Practice question · Sort into groups
Sort each statement about eigenvalues and eigenvectors as true or false.
Groups: True · False
- A negative eigenvalue flips the eigenvector's direction
- Every vector is an eigenvector of every matrix
- The eigenvalue is the scale factor along the eigenvector
- An eigenvector must be nonzero
Hints
- Recall the defining equation A v = lambda v and its 'v nonzero' condition.
- Only special directions survive as eigenvectors of a general matrix.
Show the answer
True: An eigenvector must be nonzero, The eigenvalue is the scale factor along the eigenvector, A negative eigenvalue flips the eigenvector's direction
False: Every vector is an eigenvector of every matrix
Why
Eigenvectors are nonzero by definition, the eigenvalue is the scale factor, and a negative eigenvalue reverses direction. But only special vectors are eigenvectors of a general matrix, most change direction.
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