Cayley Hamilton Theorem: Overview, Questions, Preparation

The Cayley–Hamilton Theorem is a linear algebraic concept introduced and termed after the renowned mathematicians Arthur Cayley and William Rowan Hamilton. Cayley Hamilton Theorem explains to us that a special polynomial of a given matrix is always equal to ‘Zero’. Let us understand the theorem in detail.
The theorem states that each square matrix over a commutative ring agrees with its equation. The commutative ring includes real or complex fields.

For instance, if A is provided as n x n matrix and ln is n x n identity matrix,

Then, the distinctive polynomial of A is expressed as p (x) = det (xln – A)

Here, det refers to determinant operation; while for the ‘scalar element of the base ring’, the variable is expressed as x. The matrix entries are both linear or constant polynomials in x; the det is also n-th monic polynomial in x.

The Cayley–Hamilton theorem states that by substituting matrix, A for x in the polynomial p (x) = det (xln – A) will result in ‘Zero’ matrices, i.e., p (A) = 0
The theorem states that in the n x n matrix, when A is substituted by det (tI – A), a monic polynomial of degree n. In such case, the power of A, found by replacing the power of x (identified by recurrent matrix multiplication), the constant term of p (x), results in a power of A^0 (power stands for identity matrix)
In this theorem, A^n is allowed to be articulated as a ‘linear combination’ with A as lower matrix power. If the ring is presumed to be the field, the Cayley–Hamilton theorem would be equivalent to the statement stating that the smallest polynomial of a square matrix will be divided by its characteristic polynomial.

For example,

Consider that in an n × n matrix A over ℂ and the polynomial
p (x) = det (xln – A)

with the characteristic equation
p (A) = 0

The Cayley-Hamilton theorem states that substituting the matrix A in the characteristic polynomial results in the n × n zero matrix, i.e.
p(A) = 0n

Thus, we can say that by replacing x with matrix A, the result would be equivalent to zero. Hence here the matrix A annihilates its very own characteristic equation.

What is the use of the Cayley Hamilton Theorem?

The application of the Cayley Hamilton Theorem is a basic commutative linear algebraic expression used for the computation of large matrices. 

Q: What is Cayley Hamilton Theorem?

Q: What is one of the most important applications of the Cayley Hamilton Theorem?

Q:  What is the formula of the Cayley Hamilton Theorem?

Q: Is the Cayley Hamilton Theorem applicable to all types of matrices?

Q: What does the Cayley Hamilton Theorem satisfy?