Practice question · Select all that apply
Select every TRUE statement about eigenvalues and eigenvectors.
Hints
- A negative eigenvalue reverses the direction but keeps the line, which is still allowed.
- Think about what a rotation by a quarter turn does to every direction in the plane.
Show the answer
- A. If v is an eigenvector with eigenvalue lambda, then 3v is too
- C. An eigenvalue may be negative
- D. An eigenvector must be nonzero
- E. The eigenvalues of a triangular matrix are its diagonal entries
Why
Eigenvectors are nonzero and come in lines, so any nonzero multiple works; negative eigenvalues simply flip the vector. But the quarter-turn rotation [0, -1 ; 1, 0] has characteristic equation lambda squared + 1 = 0 with no real root, it preserves no real direction at all.
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