Diagonalisation and Powers of a Matrix
When enough independent eigenvectors exist, the map is a diagonal matrix in the eigenbasis, which makes powers, long-run behaviour and matrix exponentials easy.
What a learner can do afterwards
- Diagonalise a matrix and state the two factors involved
- Compute a high power of a matrix through its diagonal form
- Give a matrix that cannot be diagonalised and say what fails
1 · Read
Diagonalising writes A as P D P inverse, where D holds eigenvalues on its diagonal and P holds matching eigenvectors as columns. The recipe: find all eigenvalues, find a basis of eigenvectors, stack them into P, read D off. A diagonal matrix shows it plainly: diag(4, 7) has eigenvalues 4 and 7.
Powers then collapse beautifully: A to the k equals P D to the k P inverse, with D to the k just raising diagonal entries. With D = diag(2, 3), A squared has eigenvalues 4 and 9, so its trace is 13. With eigenvalues 1 and 2, A to the 10th has trace 1 + 1024 = 1025.
This works exactly when enough independent eigenvectors exist to fill P. Distinct eigenvalues always supply them. But [[2, 1], [0, 2]] has eigenvalue 2 twice with only one eigen direction, so no eigenbasis exists and it cannot diagonalise.
An eigenspace is just the null space of A minus lambda I, so eigenvector hunting reuses your null space skills. After assembling, multiply out P D P inverse once to confirm you land back on A.
Stack eigenvectors into P, eigenvalues into D, and let the diagonal do the heavy lifting for powers.
2 · Watch
Take it off screen
Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.