The Cayley-Hamilton theorem for a linear transformation
Created almost 9 years ago, updated 5 months ago
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Theorem (Cayley-Hamilton): Every linear transformation (T: V o V) on a finite-dimensional vector space satisfies its own characteristic equation. That is, if (p_T(\lambda)) is the characteristic polynomial of (T), then: [p_T(T) = 0]
where the polynomial is evaluated at (T) (replacing (\lambda) with (T) and the constant term with that scalar times the identity transformation).
Intuition: The characteristic polynomial, when applied to the transformation itself, annihilates it. This means the minimal polynomial always divides the characteristic polynomial.
Example: For (A = egin{pmatrix} 1 & 2 \ 3 & 0 \end{pmatrix}), the characteristic polynomial is (p(\lambda) = \lambda^2 - \lambda - 6). The Cayley-Hamilton theorem says (A^2 - A - 6I = 0), which can be verified directly.
**Theorem (Cayley-Hamilton):** Every linear transformation \(T: V o V\) on a finite-dimensional vector space satisfies its own characteristic equation. That is, if \(p_T(\lambda)\) is the characteristic polynomial of \(T\), then:
\[p_T(T) = 0\]
where the polynomial is evaluated at \(T\) (replacing \(\lambda\) with \(T\) and the constant term with that scalar times the identity transformation).
**Intuition:** The characteristic polynomial, when applied to the transformation itself, annihilates it. This means the minimal polynomial always divides the characteristic polynomial.
**Example:** For \(A = egin{pmatrix} 1 & 2 \ 3 & 0 \end{pmatrix}\), the characteristic polynomial is \(p(\lambda) = \lambda^2 - \lambda - 6\). The Cayley-Hamilton theorem says \(A^2 - A - 6I = 0\), which can be verified directly.