The Cayley–Hamilton Theorem

The Cayley–Hamilton theorem is true for only square matrices. A matrix with an equal number of horizontal rows and vertical columns and can be represented using the order m x m is known as a square matrix. Square matrices are an entirely different concept of matrices; many different concepts like orthogonal matrix, symmetric and skew-symmetric matrix, etc., are developed using square matrix. One such theorem is the Cayley – Hamilton theorem.

History:

This theorem is designed for linear algebra and was theorized by two great mathematicians of the 17th century. Arthur Cayley introduced the basic idea, and he was assisted in his work by Willian Rowan Hamilton in England. Cayley found the famous British School of Pure Mathematics for young minds to contribute to Math.

Frobenius validated the theory's result for the first time in 1878. William Rowan Hamilton unintentionally made the first record of the Cayley-Hamilton theorem in his book "Lectures on Quaternions." In 1858, Arthur Cayley brought Hamilton's theory to the matrix world. Cayley's application of the same idea in the matrix of order 3 x 3 yields the predicted results. Thus, the Cayley-Hamilton theorem was so found.

Theorem:

The theorem states that a square matrix is satisfied by its characteristic equation when the matrix presides over a commutative circle, like the complex and real fields. This theorem paved a new way for solving equations using square matrices.

This theorem is coherently formulated as f(x) = determinant (xIa – B), where B is the square matrix with order a x a and Ia is the identity matrix of B. (xIa – B) is known as the characteristic polynomial of the square matrix. 'x' is nothing but a variable. Since we are dealing with linear polynomials, the determinant also becomes an nth order monic equation in x, and f(x) is the polynomial.

Characteristic Polynomial: The characteristic polynomial of any square matrix in a linear algebraic system is invariant polynomial under matrix similarity. A characteristic polynomial has eigenvalues as its roots. Its coefficients include the determinant and the trace of the matrix.

The Cayley–Hamilton theorem states that substituting the matrix B for x in polynomial, f(x) = det (xIn – B), results in the zero matrices, such as: p(B) = 0

The characteristic polynomial det (tI – B), which is a monic polynomial of degree n, solves a 'k x k' matrix B. The powers of B obtained by substituting powers of x are determined by recurring matrix multiplication; the constant term of p(x) gives a multiple of the power B0, where the power is specified as an identity matrix.

Example:

Now that we have all the knowledge about the Cayley-Hamilton theorem let us look at a simple example related to it. Let us assume, B is a square matrix of order 3×3. Therefore, the characteristic equation according to Cayley-Hamilton Theorem would be:

=│B - kI│ = 0

= k³ + C₁k₂ + C₂k + C₃I

= Substituting k with B, we get

= B³ +B₁A₂ + B₂A + B₃I