Practice question · Put in order
Order the steps of diagonalizing an n by n matrix A.
- Put those eigenvectors as the columns of P and the matching eigenvalues on the diagonal of D
- Check that the eigenvectors collected together number n and are independent
- Find the eigenvalues from the characteristic equation
- Find a basis of the eigenspace for each eigenvalue
- Write A = P D P inverse
Hints
- Eigenvalues come before eigenvectors, and the count check comes before building anything.
- The columns of P and the diagonal entries of D must correspond in the same order.
Show the answer
- Find the eigenvalues from the characteristic equation
- Find a basis of the eigenspace for each eigenvalue
- Check that the eigenvectors collected together number n and are independent
- Put those eigenvectors as the columns of P and the matching eigenvalues on the diagonal of D
- Write A = P D P inverse
Why
The check at step 3 is the one people skip: if the eigenvectors number fewer than n, no P exists and the process must stop. When they do number n, the j-th column of P must be an eigenvector for the j-th diagonal entry of D, misaligning them silently produces a wrong factorisation.
Practise Diagonalization
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