Diagonalisation
Objective. Write A=PDP⁻¹ and use it to calculate powers.
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- 2. Worked example
- 3. Practice
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Learn: the key idea
Diagonalisation expresses a matrix M as M = PDP⁻¹, where D is a diagonal matrix of eigenvalues and P has the corresponding eigenvectors as its columns, provided M has enough linearly independent eigenvectors. This is useful because powers of M become easy to compute: Mⁿ = PDⁿP⁻¹, and Dⁿ is simply the diagonal matrix with each eigenvalue raised to the power n. For example, if M has eigenvalues 2 and 3 with eigenvectors (1,1) and (1,−1), then M=PDP⁻¹ with D=diag(2,3) lets M¹⁰ be computed without multiplying M by itself ten times.
Worked example
The matrix M = [[3,0],[0,5]] is already diagonal with eigenvalues 3 and 5. Use diagonalisation to find M³ quickly.
- 1Since M is diagonal, D=M, with eigenvalues 3 and 5 on the diagonal
- 2For a diagonal matrix, Mⁿ is simply each diagonal entry raised to the power n
- 3Compute 3³=27 and 5³=125
- 4So M³ = [[27,0],[0,125]]
Answer: M³ = [[27,0],[0,125]]
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Start practisingCommon mistake. Students assemble P with eigenvectors in a different order than the corresponding eigenvalues in D, producing a completely incorrect diagonalisation. The order of eigenvectors in P's columns must exactly match the order of eigenvalues in D.
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