Frequently Asked Questions
When is a matrix diagonalizable?
When it has as many linearly independent eigenvectors as its dimension. Distinct eigenvalues are sufficient; repeated eigenvalues require a check on the eigenspace dimension.
How is diagonalization used?
Powers of A become A to the n equals P D to the n times P inverse - just raise each diagonal entry. Useful for solving recurrences, computing matrix exponentials, and analyzing dynamical systems.
What is a defective matrix?
A matrix that is not diagonalizable: it has a repeated eigenvalue with fewer eigenvectors than the multiplicity. Example: [[1, 1], [0, 1]].
How do you diagonalize a matrix step by step?
Find the eigenvalues from det(A − λI) = 0, find an eigenvector for each, place the eigenvectors as columns of P and the eigenvalues on the diagonal of D. Then A = PDP⁻¹, provided you have enough independent eigenvectors.
Can every matrix be diagonalized?
No. A matrix is diagonalizable only if it has a full set of linearly independent eigenvectors. A repeated eigenvalue that lacks enough eigenvectors makes the matrix defective; some real matrices also diagonalize only over the complex numbers.
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