Concept 06 / 09Matrices and Determinants
Characteristic Equation, Cayley-Hamilton Theorem and Powers of Matrices
Eigenvalues as roots of |A - lambda I| = 0, the trace and determinant as their sum and product, the Cayley-Hamilton theorem, and using it to find inverses and high powers of a matrix without repeated multiplication.
For a square matrix , a non-zero vector is an eigenvector with eigenvalue if : the transformation does not turn , it only scales it. Rewriting, with , which is a homogeneous system with a non-trivial solution. By the previous concept, this happens exactly when
.
This polynomial equation in is the characteristic equation of , and its roots are the eigenvalues. For a matrix it reads , where the trace is the sum of the diagonal entries. For a matrix, .
By Vieta's formulas, the sum of the eigenvalues equals the trace and the product equals the determinant. So a matrix is singular exactly when is an eigenvalue. Other quick facts: the eigenvalues of a triangular matrix are its diagonal entries; and have the same eigenvalues; if is an eigenvalue of , then is an eigenvalue of , of , and of . Real symmetric matrices have only real eigenvalues.
The figure shows : the characteristic equation is , so and . Vectors along are stretched by , vectors along are unchanged, and any other vector is turned towards the direction.
The Cayley-Hamilton theorem: every square matrix satisfies its own characteristic equation. If , then . For matrices, , which you can verify directly by multiplying out. A tempting but wrong 'proof' is to substitute into and get ; this confuses the number with the zero matrix. The real proof uses the adjoint of .
Uses of Cayley-Hamilton.
Inverse: if , the constant term is non-zero, so rearrange and factor out . For : .
High powers: every power can be reduced to a combination of . For , . One way to find : the same relation holds for the eigenvalues, for each eigenvalue, giving two linear equations (when the eigenvalues are distinct).
Special cases: if (idempotent) then for all ; if (involutory) then alternates between and ; if then .
Key Takeaways:
- Eigenvalues: roots of . Sum , product .
- characteristic equation: .
- Cayley-Hamilton: for .
- for .
- singular is an eigenvalue.
- Triangular matrix: eigenvalues are the diagonal entries.
- If , then .
- Pitfall: 'proving' Cayley-Hamilton by substituting into a determinant.
Brain-Teaser 1: For , express as and find .
Solution: and , so Cayley-Hamilton gives . Then . For the inverse, , so and . Check against the formula: , which is the same. Repeatedly replacing reduces any power to a linear expression.
Brain-Teaser 2: is a matrix with eigenvalues . Find , , and .
Solution: . The eigenvalues of are , so . The eigenvalues of are , that is , whose product is . And . Working with eigenvalues turns questions about complicated matrices into arithmetic on three numbers.
Brain-Teaser 3: If , find and the eigenvalues of .
Solution: Every entry of is a row of ones dotted with a column of ones, which is , so . Then by induction: . For eigenvalues: has rank , so , and is an eigenvalue of multiplicity (any vector with entries summing to is sent to ). The trace is , so the remaining eigenvalue is , with eigenvector . The relation also says that every eigenvalue satisfies , so , which is consistent.
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