Cayley-Hamilton Theorem - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

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 Cayley-Hamilton theorem is quite an important theorem used in matrix theory. This theorem is named after its founder mathematicians Arthur Cayley and William Hamilton.

Explanation: 
Let us consider that A be an n×nn×n square matrix and if its characteristic polynomial is defined as:
P(λ)=|A−λ In
Where, I is the identity matrix of same order as A .

Then, according to Cayley-Hamilton theorem:
P(A)  = O 
Where, O  represents the zero matrix of same order as A .
We can say that if we replace λ by matrix A , then the relation would be equal to zero. Hence matrix A annihilates its own characteristic equation. 

 

Proof

Let us assume a square matrix A of dimension n×nn×n. If P(λ) be its characteristic polynomial, then by the definition of characteristic polynomial:

Also, let us suppose that Q(λ) be the adjoint matrix of A−λI, such that:

Cayley-Hamilton Theorem - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

 

We have the formula:
(adjA)A = (detA)I

Cayley-Hamilton Theorem - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Cayley-Hamilton Theorem - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Also, we have:

Cayley-Hamilton Theorem - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

On comparing (1) and (2), we get:

Cayley-Hamilton Theorem - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

On multiplying ascending powers of A in each equation,

Cayley-Hamilton Theorem - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

On adding all the equations together, Everything on left hand side cancels out and we obtain,

Cayley-Hamilton Theorem - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Hence, the statement of Cayley Hamilton theorem is proved.

 

Examples

The examples based on Cayley Hamilton theorem are illustrated below:
Example 1 :  Prove Cayley Hamilton theorem for the following matrix?

Cayley-Hamilton Theorem - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Solution:  Let A = Cayley-Hamilton Theorem - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Let us find characteristic polynomial of given matrix.

Cayley-Hamilton Theorem - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Cayley-Hamilton Theorem - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

In order to prove the statement of Cayley Hamilton theorem for A, we need to show that:

Cayley-Hamilton Theorem - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Cayley-Hamilton Theorem - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

P(A) = O, hence Cayley Hamilton theorem for given matrix A is proved.


Example 2 : If Cayley Hamilton theorem holds for the matrix, then find its inverse.

Cayley-Hamilton Theorem - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Its characteristic polynomial is -

Cayley-Hamilton Theorem - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

According to Cayley Hamilton theorem -

Cayley-Hamilton Theorem - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Cayley-Hamilton Theorem - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Therefore, equation (1) becomes - 

 

Cayley-Hamilton Theorem - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Cayley-Hamilton Theorem - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev


Applications

Cayley Hamilton theorem is widely applicable in many fields not only related to mathematics, but in other scientific fields too. This theorem is used all over in linear algebra. One can easily find inverse of a matrix using Cayley Hamilton theorem. It also plays an important role in solving ordinary differential equations. This theorem is quite useful in physics also. Cayley Hamilton theorem plays a vital role in computer programming and coding. In a newer subject - Rheology, where behaviour of material is studied, this theorem is used to determine the equations that illustrate nature of materials.

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