Matrix Exponential Theorem: Proving Formally with Cayley-Hamilton

In summary, the theorem states that for a square matrix A of size n x n, the matrix exponential exp(At) can be expressed as a linear combination of powers of A and t. The coefficients in the linear combination are functions of t and can be written as a polynomial r(lambda). This theorem is proved using the Jordon decomposition theorem, which reduces the problem to one on Jordon blocks. The Cayley-Hamilton theorem is also mentioned as a rigorous proof for this theorem. However, the Jordon decomposition theorem is used in the proof.
  • #1
matematikawan
338
0
Theorem
Let A be a square matrix nXn then exp(At) can be written as

[tex]
exp(At)=\alpha_{n-1}A^{n-1}t^{n-1} + \alpha_{n-2}A^{n-2}t^{n-2} + ... + \alpha_1At + \alpha_0 I
[/tex]

where [tex]\alpha_0 , \alpha_1 , ... , \alpha_{n-1}[/tex] are functions of t.
Let define

[tex] r(\lambda)=\alpha_{n-1}\lambda^{n-1} + \alpha_{n-2}\lambda^{n-2} + ... + \alpha_1\lambda + \alpha_0 [/tex].

If [tex]\lambda_i[/tex] is an eigenvalue of At with multiplicity k, then

[tex] e^{\lambda_i }= r(\lambda_i) [/tex]
[tex] e^{\lambda_i} = \frac{dr}{d\lambda}|_{\lambda=\lambda_i} [/tex]
etc

Does anyone know any reference where it gives a proof for this theorem? I only know how to prove this theorem intuitively using the Cayley-Hamilton theorem. I need a formal proof. The book (Schaum Outline Series) that I got it only state the theorem.

This theorem will allowed me later to solve system of linear differential equations.

Please help.
 
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  • #2
The Cayley-Hamilton theorem is rigorous. The difficulty is that the a_i are all seies which you must show converge. Do you know the Jordon decomposition theorem?

Jordon decomposition theorem
Let A be a linear operator over the field C (complex numbers can be generalizes)
The exist an invertable linear operator S (made up of the generalized eigenvectors of A) such that SA=BS where B is a direct sum of Jordan blocks.

let ' denot inverse
now we have reduced the problem to one on Jordon blocks
J=aI+H
exp(J)=exp(a)exp(H)
H^n=0 so the infinite sum has all higher terms zero
QED
 
  • #3
We are computing the matrix exponential in an undergraduate engineering mathematics class. As such we never came across the Jordon decomposition theorem before. No wonder I have difficulty in finding the literatures for the proof.
Now there is a hint. I will again search the literatures or ask one of our professor of algebra.
Thanks lurflurf.


I will come back to this thread later, especially because I don't understand a word in the proof. :cry:
lurflurf said:
let ' denot inverse
now we have reduced the problem to one on Jordon blocks
J=aI+H
exp(J)=exp(a)exp(H)
H^n=0 so the infinite sum has all higher terms zero
QED

If only someone could proved the theorem using the Cayley-Hamilton theorem !:smile:
 
  • #4
That professor of algebra is on leave. It's chinese new year holiday.

:smile: Searching the internet, I found the proof that I wanted.
http://web.mit.edu/2.151/www/Handouts/CayleyHamilton.pdf

Along the way I get in love with that jordan decomposition. :!) Still reading.
In fact jordan decomposition may refer to quite different concept
http://en.wikipedia.org/wiki/Jordan_decomposition

The one given by lurflurf is Jordan normal form.

I think computing matrix exponential using Jordan matrix ( ref: http://en.wikipedia.org/wiki/Matrix_exponential ) is less efficient compare to that using the theorem that I had stated.
 
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Related to Matrix Exponential Theorem: Proving Formally with Cayley-Hamilton

1. What is the Matrix Exponential Theorem?

The Matrix Exponential Theorem states that any square matrix can be expressed as a linear combination of powers of its eigenvalues. This theorem is useful in solving differential equations and in understanding the behavior of linear systems.

2. How is the Matrix Exponential Theorem proved formally?

The Matrix Exponential Theorem can be proved using the Cayley-Hamilton Theorem, which states that every square matrix satisfies its own characteristic equation. By substituting the matrix into its characteristic equation, we can show that it can be expressed as a linear combination of its eigenvalues.

3. What is the significance of the Cayley-Hamilton Theorem in proving the Matrix Exponential Theorem?

The Cayley-Hamilton Theorem is significant in proving the Matrix Exponential Theorem because it allows us to express a matrix as a linear combination of its eigenvalues, which is a key component of the Matrix Exponential Theorem.

4. Why is the Matrix Exponential Theorem important in mathematics?

The Matrix Exponential Theorem is important in mathematics because it provides a powerful tool for solving differential equations and understanding the behavior of linear systems. It also has applications in fields such as physics, engineering, and computer science.

5. Are there any limitations or assumptions to the Matrix Exponential Theorem?

The Matrix Exponential Theorem only applies to square matrices and assumes that the matrix has distinct eigenvalues. It also does not hold for all types of matrices, such as non-diagonalizable matrices. Additionally, the proof of the theorem relies on the Cayley-Hamilton Theorem, which has its own limitations and assumptions.

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