Raising a bunch of matrices to a power

In summary, under certain conditions, it is possible to simplify (ABC)^5 to expand it into (A^5)(B^5)(C^5). The fastest way to solve (ABC)^5 is to multiply ABC out and then diagonalize it, or to write it in terms of A^5, B^5, C^5, and the commutators [A, B], [A, C], and [B, C]. However, this method may not always be helpful.
  • #1
sjeddie
18
0
Let A, B, and C be nxn matrices,
I'm wondering
1. is it possible to simplify (ABC)^5 to expand it ( maybe into something like (A^5)(B^5)(C^5) )
2. what's the fastest way of solving (ABC)^5? I'm thinking actually multiply ABC out, then diagonalize it. Is there a faster way?
 
Physics news on Phys.org
  • #2
sjeddie said:
Let A, B, and C be nxn matrices,
I'm wondering
1. is it possible to simplify (ABC)^5 to expand it ( maybe into something like (A^5)(B^5)(C^5) )
Under certain conditions. If A, B and C commute among themselves (that is, AB = BA, AC = CA and BC = CB) then this is possible.

sjeddie said:
2. what's the fastest way of solving (ABC)^5? I'm thinking actually multiply ABC out, then diagonalize it. Is there a faster way?

I'm thinking the same thing. Alternatively you could try writing down an expression in terms of A^5, B^5, C^5 and the commutators [A, B] = AB - BA, [A, C] = AC - CA and [B, C] = BC - CB, e.g.
(A B C)^2 = A B C A B C
... = A B A C B C + A B [C, A] B C
... = A^2 B C B C + A [B, A] C B C + A B [C, A] B C
... = A^2 B^2 C^2 - A^2 B [B, C] C - A [A, B] C B C - A B [A, C] B C

But that is in general ugly and not very helpful.
 
  • #3
Thank you CompuChip
 

Related to Raising a bunch of matrices to a power

1. What does it mean to raise a matrix to a power?

When a matrix is raised to a power, it means that the matrix is multiplied by itself a certain number of times. The power indicates the number of times the matrix is multiplied by itself.

2. Can all matrices be raised to a power?

No, not all matrices can be raised to a power. Only square matrices, which have an equal number of rows and columns, can be raised to a power. This is because matrix multiplication can only be performed on matrices with compatible dimensions.

3. How is matrix multiplication used to raise a matrix to a power?

To raise a matrix to a power, the matrix is multiplied by itself the number of times indicated by the power. For example, to find the matrix A raised to the power of 3, it would be multiplied by itself two times: A x A x A.

4. What is the result of raising a matrix to the power of 0?

Raising a matrix to the power of 0 results in an identity matrix, which is a square matrix with 1s along the main diagonal and 0s everywhere else. This is because any matrix multiplied by the identity matrix results in the original matrix.

5. How does raising a matrix to a negative power work?

Raising a matrix to a negative power is equivalent to finding the inverse of the matrix and then raising it to the positive power. Inverse matrices are used to "undo" matrix multiplication, so raising a matrix to a negative power undoes the matrix multiplication that would occur when raising it to a positive power.

Similar threads

  • Linear and Abstract Algebra
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
523
  • Precalculus Mathematics Homework Help
2
Replies
69
Views
4K
  • Special and General Relativity
Replies
12
Views
2K
Replies
6
Views
918
  • Calculus and Beyond Homework Help
Replies
2
Views
2K
  • Programming and Computer Science
Replies
11
Views
681
  • Math Proof Training and Practice
3
Replies
80
Views
5K
  • Linear and Abstract Algebra
Replies
2
Views
1K
Back
Top