Prooving the Cayley-Hamilton Theorem

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In summary, the Cayley-Hamilton Theorem states that every matrix is a zero of its characteristic equation and can be proven by comparing coefficients and adding. The proof involves using the non-commutative root/factor theorem and can be found in A. Adrian Albert's book "Fundamental Concepts of Higher Algebra."
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Vuldoraq
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Proving the Cayley-Hamilton Theorem

Hey all,

I'm revising for my linear algebra exam, which is next week, and I got up to the Cayley-Hamilton theorem, but I am stuck on the final leap in the proof. Here is what I understand so far,

Theorem:

Every matrix is a zero of it's characteristic equation[tex][P_{A}(\lambda)=det(A-\lambda I)][/tex],

For [tex]\\\\\\\\\\\\\ A \in M_{n}(k)[/tex] [tex] P_{A}(A)=0[/tex]


Proof:

[tex]
Let\ P_{A}(\lambda)=det(A-\lambda I)=\lambda^{n}+a_{n-1}\lambda^{n-1}+\ldots+a_{1}\lambda+a_{0}[/tex]

and consider,

[tex]\phi(\lambda)=adj(A-\lambda I)=B_{n-1}\lambda^{n-1}+ldots+B_{1}\lambda+B_{0}[/tex]

where [tex] B_{i} \in M_{n}(k)[/tex]

Given that for any [tex]C \in M_{n}(k); \\ C*adj(C)=det(C)*I.[/tex]

So by letting [tex] C=A-\lambda I[/tex] we have,

[tex](A- \lambda I)*\phi(\lambda)=det(A-\lambda I)I=P_{A}(\lambda)I[/tex]

Expanding we have,

[tex](A- \lambda I)*(B_{n-1}\lambda^{n-1}+ldots+B_{1}\lambda+B_{0})=(\lambda^{n}+a_{n-1}\lambda^{n-1}+ldots+a_{1}\lambda+a_{0})*I[/tex]

The next step has me in tears, my book says compare coefficients and add but I can't see how you would compare these? Please can anyone help me to complete this?
 
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  • #2
Multiply the LHS and then compare for each ##i=0,\ldots,n## the coefficients of ##\lambda^i##. E.g. for ##i=n## we get ##- B_{n-1}=I##, for ##i=n-1## we have ##AB_{n-1}-B_{n-2}=a_{n-1}I##. With ##B_{n-1}=-I## from the previous step, we get ##B_{n-2}=-A-a_{n-1}I## and so on.
 
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  • #3
Thanks for the reply @fresh_42! I was quite surprised to see one after 11 years, but it's appreciated.

Totally forgot this forum existed, will see if I can brush up on my physics and maths and get more involved.
 
  • #4
In fact Cayley Hamilton follows immediately from the next to last equation in post # 1, viewed as an equation between polynomials with matrix coefficients, if you know the non commutative root/factor theorem, namely that the fact (A-t) is a (left) factor of P(t) implies that t=A is a (left root, hence also a) root of P(t). This proof occurs in Fundamental Concepts of higher algebra, by A. Adrian Albert, p.84.
 

1. What is the Cayley-Hamilton Theorem?

The Cayley-Hamilton Theorem is a fundamental theorem in linear algebra that states any square matrix satisfies its own characteristic equation. In other words, a matrix can be used to evaluate its own polynomial equation.

2. Why is the Cayley-Hamilton Theorem important?

The Cayley-Hamilton Theorem has many applications in mathematics, physics, and engineering. It is used to solve systems of linear equations, find eigenvalues and eigenvectors, and prove other theorems. It also has implications in quantum mechanics and control theory.

3. How was the Cayley-Hamilton Theorem discovered?

The theorem was first discovered by mathematicians Arthur Cayley and William Rowan Hamilton in the mid-19th century. They both independently proved the theorem and it was later named after them. However, the idea of the theorem can be traced back to Leonhard Euler's work in the 18th century.

4. Can the Cayley-Hamilton Theorem be proven?

Yes, the Cayley-Hamilton Theorem can be proven using various methods such as matrix algebra, linear transformations, and eigenvalues. The proof is a fundamental concept in linear algebra and is often taught in undergraduate courses in mathematics and physics.

5. What are the implications of the Cayley-Hamilton Theorem in real-world applications?

The Cayley-Hamilton Theorem has many applications in various fields such as image processing, signal processing, data compression, and control systems. It is also used in computer graphics, cryptography, and optimization problems. Its applications are widespread and have greatly influenced modern technology and scientific advancements.

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