Practice question · Select all that apply
Select every TRUE statement about diagonalization.
Hints
- Two options make an absolute claim; check each against the identity matrix and against a defective matrix.
- The identity matrix repeats its eigenvalue and is already diagonal.
Show the answer
- B. Having n distinct eigenvalues guarantees diagonalizability
- C. In A = P D P inverse the columns of P are eigenvectors of A
- D. The diagonal entries of D are the eigenvalues of A
Why
Distinct eigenvalues force independence and hence diagonalizability, and P and D are built exactly from eigenvectors and eigenvalues. But some matrices are defective, so not every matrix is diagonalizable, and the identity shows repetition alone is harmless.
Practise Diagonalization
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