# Raising a bunch of matrices to a power

• sjeddie
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.
sjeddie
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?

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.

Thank you CompuChip

## 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.

• Linear and Abstract Algebra
Replies
5
Views
1K
• Calculus and Beyond Homework Help
Replies
2
Views
523
• Precalculus Mathematics Homework Help
Replies
69
Views
4K
• Special and General Relativity
Replies
12
Views
2K
• Quantum Physics
Replies
6
Views
918
• Calculus and Beyond Homework Help
Replies
2
Views
2K
• General Math
Replies
7
Views
1K
• Programming and Computer Science
Replies
11
Views
681
• Math Proof Training and Practice
Replies
80
Views
5K
• Linear and Abstract Algebra
Replies
2
Views
1K