Diagonalisation
If A has enough independent eigenvectors, then A = P D P^(-1). The columns of P are the eigenvectors and D has the eigenvalues on its diagonal.
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Analogy
Viewing a problem from the right angle, so the matrix just stretches each axis separately.
Example
P = [[1,1],[1,-2]], D = [[5,0],[0,2]].P inverse = [[2/3,1/3],[1/3,-1/3]].P D P inverse = [[4,1],[2,3]].Check first row: 4 and 1.
In practice
Makes matrix powers and differential equations easy to solve.
Practice
Flashcards
- Write the diagonalisation formula.A = P D P^(-1).
- How do you compute A^n using diagonalisation?A^n = P D^n P^(-1).
Interview questions
- Explain how diagonalisation helps compute high powers of a matrix.
- A = P D P^(-1)
- A^n = P D^n P^(-1)
- D^n only needs each diagonal entry raised to n
- Largest eigenvalue dominates the long run
Questions people ask
Which matrices can be diagonalised?
Those with a full set of independent eigenvectors, meaning n of them for an n by n matrix. A matrix with n different eigenvalues always works. A matrix like [[1,1],[0,1]] has only one eigenvector direction, so it cannot be diagonalised.
Why is the diagonal form useful?
A diagonal matrix is easy to work with: to square it, just square each diagonal entry. After changing basis to the eigenvectors, a complicated matrix acts like a simple diagonal one. You then switch back to the original basis at the end.