Matrix Power Calculator

Matrix power Calculator 

Matrix power calculator can find the exponent of the matrix up to 6th power. It gives calculations done at each step. You can enter a 2 by 2 to a 5 by 5 matrix in this power of a matrix calculator. 

What is matrix power?

Matrices are an intelligent combination of numbers that contain a lot of information. Also, they make applying complex operations easy.

Matrix power is simply the exponent of a matrix. For a simple number like 2, exponent means multiplying 2 by itself. In matrices, you multiply the whole matrix by its equal matrix (i.e itself). 

How to find the power of a matrix?

There is no formula, just simple multiplication. You multiply the matrix by itself the n (power) number of times. 

For a smaller matrix like 2 x 2 and a small power like 2 or 3, the exponent can be found by hand. But as the matrix size increases and the power rises, the complexity becomes greater.

A matrix power calculator is helpful in such situations because a minor mistake at any point will ruin a day’s worth of hard work.

Example:

Find the 3rd power for the following matrix.

⌈3  2⌉
⌊2  1⌋

Solution:

= ⌈3  2⌉    x  ⌈3  2⌉
   ⌊2  1⌋        ⌊2  1⌋

= ⌈3(3) + 2(2)  2(3) + 1(2)⌉
   ⌊3(2) + 2(1)  2(2) + 1(1)⌋

= ⌈9 + 4   6 + 2⌉
   ⌊6 + 2   4 + 1⌋

= ⌈13   8⌉
   ⌊8     5⌋

It is power 2. Multiplying the resultant matrix with the real matrix again. 

= ⌈3  2⌉    x  ⌈13  8⌉
   ⌊2  1⌋        ⌊8    5⌋

= ⌈3(13) + 2(8)   2(13) + 1(8)⌉
   ⌊3(8) + 2(5)       2(8) + 1(5)⌋

= ⌈39 + 16   26+ 8⌉
   ⌊18 + 10  16 + 5⌋

= ⌈55   34⌉
   ⌊28   21⌋

This is the 3 times the original matrix.

How to find the power of matrices higher than 6?

If a matrix is diagonalizable i.e which can be written in the form of a diagonal matrix, we can use the eigenvalues to find its power rather easily. 

You need to understand two concepts: what is a diagonalized matrix? How is it related to its original matrix?  

• A diagonal matrix has entries only in its diagonal, all other entries are zeros. For this particular problem, while diagonalizing, the eigenvalues (of the original matrix) are the entries (non-zero ones) of the diagonal matrix. 

For example, a matrix of 2 by 2

M = ⌈1   3⌉
       ⌊7   5⌋

has eigenvalues -2 and 8 then Its diagonal matrix is:

D = ⌈-2   0⌉
       ⌊0    8⌋

• A matrix M is related to its diagonal matrix as

M = E.D.E^-1  

Here D is the diagonal matrix of M. While E is such a matrix that is formed if you place eigenvectors of matrix M, as columns. E^-1 is its inverse matrix.

So how does it help to find the matrix power?

Say you want to find the 12th power of a matrix. If you take this power for the entries of the diagonal matrix and use that new matrix, say D¹² in the equation mentioned above, you will have the 12th power of the original matrix.

For instance, take the previous example of the matrix M we diagonalized. If you want to find its 12th power, take the 12th power of the entries of its diagonal matrix i.e

= ⌈-2   0⌉
   ⌊0    8⌋

Taking power

D12 = ⌈-212   0⌉
         ⌊0    812⌋

Now, find the eigenvectors and make the matrix E. The vectors are:

V1 = (-1,1)
V2 = (0.43, 1)

The new matrix E is 

= ⌈-1   0.43⌉
   ⌊1         1⌋

Find its inverse, the E^-1 matrix. Use the inverse matrix calculator to save time. After that multiply the matrix E with matrix D¹² and lastly with E^-1. You have the required answer.

Calculations are still complex using this method but the procedure is short. So that’s an edge it got over simple multiplication. To sum it all up in a few steps:

1. Find eigenvalues and eigenvectors.
2. Use eigenvalues to make a diagonal matrix.
3. Use eigenvectors to make a new matrix (name it whatever you want) and find its inverse.
4. Take the required exponent for the entries of the diagonal matrix. 
5. Multiply the new diagonal matrix with the “eigenvectors-matrix”.
6. Lastly, multiply it with the inverse of the “eigenvector-matrix”.