Maths › Further Pure 2 › Diagonalisation and the Cayley-Hamilton theorem
Diagonalisation and the Cayley-Hamilton theorem
Change coordinates so the eigenvectors become the axes and the matrix turns diagonal, making high powers trivial. And every matrix, it turns out, satisfies its own characteristic equation.
Builds on Eigenvalues and eigenvectors and Determinants and inverses.
IN THIS TOPIC
- Build P from eigenvectors and D from eigenvalues so that P⁻¹MP = D.
- Use diagonalisation to compute high powers of a matrix.
- Apply the Cayley-Hamilton theorem to find powers and inverses.
WHAT YOU PROBABLY THINK
Finding the tenth power of a matrix means multiplying it out ten times.
New axes, diagonal matrix
Set the eigenvectors as the columns of P and the matching eigenvalues down the diagonal of D. Then MP = PD, and since P is invertible whenever the eigenvectors are independent:
In eigenvector coordinates the transformation is nothing but a stretch along each axis. Powers become easy, because Mn = PDnP-1 and a diagonal matrix powers entry by entry. That disposes of the opening claim: ten multiplications become two, whatever the exponent.
WORKED EXAMPLE
A tenth power without ten multiplications
For the matrix with rows (4, 1) and (2, 3), describe how to find M¹⁰.
The eigenvalues are 5 and 2, with eigenvectors (1, 1) and (1, −2), so P has columns (1, 1) and (1, −2), and D is diagonal with 5 and 2.
D¹⁰ is diagonal with 5¹⁰ = 9765625 and 2¹⁰ = 1024.
M¹⁰ = PD¹⁰P⁻¹: two multiplications and a pair of powers. The eigenvalue 5 dominates completely, so at large powers the matrix behaves as a stretch along y = x and little else.
For a symmetric matrix the eigenvectors are perpendicular, so normalising them makes P orthogonal and P⁻¹ is simply the transpose of P: the inverse costs nothing at all. That is orthogonal diagonalisation.
Every matrix solves its own equation
The Cayley-Hamilton theorem says a matrix satisfies its own characteristic equation. For a 2 × 2 with trace t and determinant d, that means M² − tM + dI = 0. Rearranging gives M² in terms of M and I, so every higher power collapses down to a linear combination of the two.
WORKED EXAMPLE
Two uses of one identity
For the matrix with rows (4, 1) and (2, 3), verify Cayley-Hamilton and use it to find M⁻¹.
M² has rows (18, 7) and (14, 11). Then M² − 7M + 10I has rows (18 − 28 + 10, 7 − 7) and (14 − 14, 11 − 21 + 10): the zero matrix, as promised.
Multiply the identity by M⁻¹: M − 7I + 10M⁻¹ = 0, so M⁻¹ = (7I − M)/10, which is (1/10) times rows (3, −1) and (−2, 4).
Checking: M times that product is the identity. The inverse came from the characteristic equation, with no adjugate anywhere.
YOUR TURN
Reducing a cube
A 2 × 2 matrix has trace 5 and determinant 6. Use Cayley-Hamilton to express M³ in terms of M and I.
Show the working
The theorem gives M² = 5M − 6I.
Multiplying by M: M³ = 5M² − 6M = 5(5M − 6I) − 6M = 19M − 30I.
Every power reduces to aM + bI for some numbers a and b, however high the exponent climbs.
THE EXAM BIT
- State P and D explicitly, with the eigenvector order matching the eigenvalue order; a mismatch invalidates everything.
- For symmetric matrices, normalise the eigenvectors so that P is orthogonal and the inverse is the transpose.
- Quote Cayley-Hamilton by name, and show the characteristic equation it comes from.
- To find an inverse from the theorem, multiply the identity through by M inverse and rearrange.
CHECK YOURSELF
A 2 × 2 matrix M has trace 6 and determinant 5. Write down the equation Cayley-Hamilton gives, and hence M⁻¹ in terms of M and I.
Show a hint
The characteristic equation is λ² − 6λ + 5 = 0.
Show the answer
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P from eigenvectors, D from eigenvalues: P⁻¹MP = D, and Mⁿ = PDⁿP⁻¹ makes powers cheap.
Cayley-Hamilton: a matrix satisfies its own characteristic equation, which yields inverses and collapses powers.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
CHECK YOUR PROGRESS
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- Build P from eigenvectors and D from eigenvalues so that P⁻¹MP = D.
- Use diagonalisation to compute high powers of a matrix.
- Apply the Cayley-Hamilton theorem to find powers and inverses.
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