27.5. EXERCISES 523
is less than the algebraic multiplicity of the eigenvalue as a root of the characteris-tic polynomial, then the matrix is called defective. It can be shown that A can bediagonalized if and only if it is nondefective. See Theorem 11.5.3.
50. The typical situation is that an n× n matrix has n distinct eigenvalues. In this case,the matrix is always nondefective. This comes from the following theorem whichyou will show in this problem.
Theorem 27.5.1 Let A be an n×n matrix and let {µ1, · · · ,µk} be distinct eigenval-ues corresponding to eigenvectors {x1, · · · ,xk}. Then this set of eigenvectors is alinearly independent set.
Do the following. If not independent, then there exist scalars ai such that
l
∑i=1
aixi = 0
in which the ai are not all zero and l is as small as possible for this to take place.Explain why al ΜΈ= 0 and why l ≥ 2. Then multiply both sides on the left by A andthen both sides on the left by µ l . Subtract and obtain a contradiction of some sort,having to do with l being as small as possible and all eigenvectors being nonzero.