# Finding A^5 for Symmetrical Matrix A

• Geronimo85
In summary, there is a method to find A^5 for a 3x3 symmetric matrix A aside from brute force multiplication, by diagonalizing A using an invertible matrix C and a diagonal matrix D. This method reduces the number of multiplications needed and involves finding the eigenvalues and eigenvectors of A.
Geronimo85
I have a 3x3 symmetrical matrix A:

1, 8^.5, 0
8^.5, 1, 8^.5
0, 8^.5, 1

that I need to find A^5 for. Is there a method aside from brute force multiplication to do so?

Geronimo85 said:
I have a 3x3 symmetrical matrix A:

1, 8^.5, 0
8^.5, 1, 8^.5
0, 8^.5, 1

that I need to find A^5 for. Is there a method aside from brute force multiplication to do so?

A fact which should be useful is that the product of symmetric matrices is symmetric itself.

Ok, how does that help?

Geronimo85 said:
Ok, how does that help?

It reduces the number of multiplications you have to do. But it's still a semi-brute multiplication method. Hope someone passes by with a better suggestion perhaps.

Since that is a symmetric matrix, it can be "diagonalized". That is there exist an invertible matrix C and a diagonal matrix D such that D= CAC-1. Then A= C-1DC. Then A5= C-1D5C. D is the matrix having the eigenvalues of A (which are easy to find) on its main diagonal, 0s off the diagonal. C is a matrix having the corresponding eigenvectors of A as columns.

## What is a symmetrical matrix?

A symmetrical matrix is a square matrix in which the elements above and below the main diagonal are mirror images of each other.

## How do you find A^5 for a symmetrical matrix A?

To find A^5 for a symmetrical matrix A, you can use the following formula: A^5 = A x A x A x A x A. This means multiplying the matrix by itself five times.

## What are the benefits of using a symmetrical matrix?

Symmetrical matrices have several benefits, including faster computation times, easier storage, and the ability to perform certain calculations more efficiently.

## Can a non-symmetrical matrix have a symmetrical A^5?

No, a non-symmetrical matrix cannot have a symmetrical A^5. In order for A^5 to be symmetrical, A must also be symmetrical.

## What are some real-world applications of symmetrical matrices?

Symmetrical matrices are commonly used in fields such as physics, engineering, and computer science for tasks such as modeling physical systems, analyzing networks, and solving optimization problems.

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