Proving Non-Diagonal Matrix Exponential is Diagonal - Ian's Problem

In summary, Gantmacher's text "The theory of matrices" (chapter VIII, section 8) provides a complete characterization of the inverse map of the matrix exponential.
  • #1
charlesworth
8
0
I'd like to do one of two things:

1: Find an example of a non-diagonal matrix whose matrix exponential (defined in terms of series) is diagonal.
2: Prove that no such examples exist.

I'm working with matrices over the complex field. My gut tells me that 2 is the way to go. I'd really appreciate any help with this.

Ian
 
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  • #2
Let [tex] \sigma_x = \left( \begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix} \right) [/tex], and consider things like [tex] e^{i \sigma_x \varphi} [/tex] where [tex] \varphi [/tex] is some angle. You should find that the matrix exponential of a non-diagonal matrix can, in fact, be diagonal.

(You can find a useful / interesting identity related to my suggestion at http://en.wikipedia.org/wiki/Pauli_matrices" [Broken]).
 
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  • #3
Or: consider [itex]A = \left( \begin{smallmatrix}0&-1\\1&0\end{smallmatrix} \right)[/itex]. Then [itex]e^{\theta A}[/itex] is a rotation by the angle [itex]\theta[/itex]. In particular, [itex]e^{n\pi A}[/itex] is diagonal for any integer [itex]n[/itex].
 
  • #4
Thanks to both of you, that's very helpful. The 2x2 examples are good, but using higher dimensional Pauli matrices is more satisfying; thanks Sando.

I found a complete answer to my question in Gantmacher's text "The theory of matrices" (chapter VIII, section 8) which gives a complete characterization of the inverse map of the matrix exponential, and in particular, said inverse map in the case of diagonal matrices.

Given a diagonal matrix D without any entries on the negative real axis, the set of "logarithms" (ie. the set {H} such that exp(H) = D) obviously includes all matrices where you just take the logarithm of each element of D, and possibly add a multiple of 2[tex]\pi[/tex]i. I had originally thought that this would be it. But as Gantmacher shows, you need to also include conjugation by anything that commutes with D. (I'm not attempting to be formal here.) Which is where the examples you provided come in.

Ian
 
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  • #5
, thank you for your question. I understand your desire to either find an example or prove the non-existence of a non-diagonal matrix whose matrix exponential is diagonal. However, I believe that proving the non-existence of such examples is the more appropriate approach in this case.

To begin, let us define what a diagonal matrix exponential means. A diagonal matrix exponential is a matrix whose elements are all zero except for the diagonal elements, which are the exponential of the corresponding elements in the original matrix. In other words, if we have a matrix A with elements a_ij, then the diagonal matrix exponential is a matrix B with elements b_ij such that b_ij = 0 for all i ≠ j and b_ii = e^(a_ii).

Now, let us assume that there exists a non-diagonal matrix A whose exponential is diagonal. This means that there exists a matrix B, with elements b_ij such that b_ij = 0 for all i ≠ j and b_ii = e^(a_ii) for all i. However, this would imply that A and B commute, since they have the same diagonal elements. This goes against the fundamental property of matrix exponentials, which states that if two matrices commute, then their exponentials also commute. In other words, if AB = BA, then e^Ae^B = e^Be^A. Since A and B are non-diagonal, they cannot commute, and therefore, their exponentials cannot be diagonal.

Furthermore, we can also consider the definition of a matrix exponential in terms of series. The series representation of the exponential function is given by e^x = 1 + x + x^2/2! + x^3/3! + ... Therefore, the series representation of a matrix exponential would be given by e^A = I + A + A^2/2! + A^3/3! + ... where I is the identity matrix. Now, if we have a non-diagonal matrix A, then its higher powers A^2, A^3, ... will also be non-diagonal. This means that the series representation of e^A will also have non-zero off-diagonal elements, and therefore, cannot be a diagonal matrix.

In conclusion, based on the fundamental properties of matrix exponentials and their series representation, it can be proven that there exists no non-diagonal matrix whose matrix exponential is diagonal. I hope this explanation helps
 

1. What is a non-diagonal matrix exponential?

A non-diagonal matrix exponential is a mathematical operation that involves raising a non-diagonal matrix to a power. It is often used in solving differential equations and has applications in various fields of science and engineering.

2. What does it mean to prove that a non-diagonal matrix exponential is diagonal?

Proving that a non-diagonal matrix exponential is diagonal means showing that the resulting matrix after the exponential operation is a diagonal matrix, where all the elements outside the main diagonal are equal to zero. This proof is important in understanding the behavior of non-diagonal matrices and their applications.

3. Why is Ian's problem of proving non-diagonal matrix exponential is diagonal significant?

Ian's problem of proving non-diagonal matrix exponential is diagonal is significant because it helps us understand the relationship between non-diagonal matrices and diagonal matrices, which have simpler properties and are easier to work with in many applications. It also has implications in solving complex mathematical problems and has practical applications in fields such as physics and engineering.

4. What are some approaches to solving Ian's problem?

There are various approaches to solving Ian's problem, including using matrix diagonalization techniques, using the Cayley-Hamilton theorem, or using the Jordan canonical form. Each approach has its own advantages and may be suitable for different types of non-diagonal matrices.

5. How is proving non-diagonal matrix exponential is diagonal useful in real-world applications?

Proving that a non-diagonal matrix exponential is diagonal has practical applications in fields such as physics, engineering, and economics. It can also be used in solving systems of differential equations and in analyzing the behavior of complex systems. In addition, understanding the properties of non-diagonal matrices can help improve algorithms and computational methods in various fields.

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