The Cayley-Hamilton Theorem

The Cayley-Hamilton theorem is a fundamental result in linear algebra that relates to the theory of matrices and their associated polynomials. It is named after the mathematicians Arthur Cayley and William Rowan Hamilton, who independently discovered the theorem in the mid-19th century.

Statement of the Theorem

The Cayley-Hamilton theorem states that every square matrix $A$ satisfies its own characteristic equation, which is given by:

where $I$ is the identity matrix and $\lambda$ is a scalar variable. This equation is also known as the characteristic polynomial of $A$.

The theorem then asserts that if we substitute $A$ for $\lambda$ in the characteristic polynomial, we obtain the zero matrix:

or equivalently:

where $n$ is the size of the matrix $A$ and $c_i$ are coefficients of the characteristic polynomial.

In other words, the Cayley-Hamilton theorem tells us that every matrix satisfies its own characteristic equation when evaluated at itself.

Applications of the Theorem

The Cayley-Hamilton theorem has important applications in several areas of mathematics and science. Here are some examples:

Matrix Diagonalization

The theorem can be used to diagonalize a matrix $A$ by showing that its characteristic polynomial factors into linear factors, each corresponding to an eigenvalue of $A$. This allows us to write $A$ as a product of its eigenvectors and eigenvalues, which simplifies many computations involving the matrix.

Minimal Polynomial

The Cayley-Hamilton theorem also implies that the characteristic polynomial is a multiple of the minimal polynomial of $A$, which is the smallest degree polynomial that annihilates $A$. This fact is useful in finding the minimal polynomial of a matrix, which is important in applications such as computing the Jordan canonical form.

Control Theory

The theorem has applications in control theory, where it can be used to prove stability of a linear system represented by a matrix. Specifically, if the characteristic polynomial has all its roots in the open left half of the complex plane, then the system is stable.

Proof of the Theorem

The proof of the Cayley-Hamilton theorem is usually done by using the adjugate (or classical adjoint) of a matrix, which is the transpose of its cofactor matrix. It can be shown that:

Using this identity, we can show that:

where the third equality follows from the fact that the adjugate of a matrix is unique up to multiplication by its determinant.

By the Cayley-Hamilton theorem, we know that $\det(A - A) = 0$. Therefore, setting $\lambda = A$ in the above equation yields:

Thus, we have shown that $A\text{adj}(A) = \det(A)I$, which implies that:

where $c_i$ are the coefficients of the characteristic polynomial. This completes the proof of the Cayley-Hamilton theorem.

Conclusion

The Cayley-Hamilton theorem is a powerful tool in linear algebra and has numerous applications in mathematics and science. Its proof involves the adjugate of a matrix and the fact that every matrix satisfies its own characteristic equation. By understanding this theorem, we gain a deeper insight into the properties of matrices and their associated polynomials.

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ケイリー・ハミルトンの定理[JA]