Diagonalisation
Objective. Characterise and apply diagonalisable operators.
- 1. Learn
- 2. Worked example
- 3. Practice
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Learn: the key idea
Diagonalisation expresses a square matrix A as PDP⁻¹, where D is diagonal containing eigenvalues and P's columns are corresponding linearly independent eigenvectors. A matrix is diagonalisable precisely when the geometric multiplicity of each eigenvalue equals its algebraic multiplicity, giving enough independent eigenvectors to form a basis. For A = [[2,0],[0,3]], A is already diagonal, so D = A and P = I. For A = [[3,1],[0,3]], the single eigenvalue 3 has only a one-dimensional eigenspace, so A is not diagonalisable despite having a repeated eigenvalue.
Worked example
Diagonalise the matrix A = [[1,0],[0,4]] if possible, and state P and D.
- 1Since A is already diagonal, its eigenvalues are 1 and 4 read directly off the diagonal.
- 2Eigenvectors are the standard basis vectors (1,0) and (0,1), already linearly independent.
- 3Set P = [[1,0],[0,1]] = I, the identity matrix, using these eigenvectors as columns.
- 4Then D = P⁻¹AP = A itself, since P is the identity, confirming A = PDP⁻¹.
Answer: P = I, D = diag(1, 4)
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Start practisingCommon mistake. Students believe every matrix with real eigenvalues is diagonalisable; defective matrices like [[5,1],[0,5]] have a repeated eigenvalue but insufficient independent eigenvectors, so they cannot be diagonalised.
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