In this explainer, we will learn how to use the matrix multiplication to determine the square and cube of a square matrix.
There are many matrix operations that are very similar to the well-known operations from conventional algebra, such as addition, subtraction, and scaling. Additionally, although matrix multiplication is fundamentally more complex than its conventional counterpart, it does still, to some extent, mirror some of the algebraic properties of the original.
One operation that is central to both conventional algebra and algebra using matrices is that of exponentiation, which is usually referred to as taking the power of a number or matrix. In conventional algebra, it is possible to take almost any number and raise it to a power , giving . With the exception of taking zero to a negative power, it does not matter whether or is zero, nonzero, integer, noninteger, rational, irrational, or complex as the output can always be calculated. The same is not true when working with matrices, where a matrix cannot always be exponentiated. In order to best outline these potential complications, let us first define the simplest form of matrix exponentiation: squaring a matrix.
Definition: Square of a Matrix
If is a square matrix, is defined by
In other words, just like for the exponentiation of numbers (i.e., ), the square is obtained by multiplying the matrix by itself.
As one might notice, the most basic requirement for matrix exponentiation to be defined is that must be square. This is because, for two general matrices and , the matrix multiplication is only well defined if there is the same number of columns in as there are rows in . If has order and has order , then is well defined and has order . If we were only to consider the matrix and attempt to complete the matrix multiplication , then we would be attempting to multiply a matrix with order by another matrix with order . This can only be well defined if , meaning that has to be a matrix with order (in other words square). The order of is therefore identical to the original matrix .
There are also other restrictions on taking the powers of matrices that do not exist for real numbers. For instance, unlike with regular numbers, we have no way of defining what is, and the negative power of a matrix is much more difficult to calculate. Furthermore, the usual laws of exponentiation do not necessarily extend to matrices in the same way as they do for numbers, which we will investigate later in this explainer.
For now, let us demonstrate how squaring a matrix works in a simple, nontrivial case. We define the matrix
To calculate matrix , we are multiplying the matrix by itself. In other words, we have
As expected, this multiplication is well defined, since we have a matrix multiplied by a matrix. It now remains to complete the matrix multiplication, which we can do for each entry by multiplying the elements in row of the left matrix by the elements in column of the right matrix and by summing them up. We demonstrate this process below:
Now that all entries have been computed, we can write that
Let us now consider an example where we can apply this technique of squaring a matrix to solve a problem.
Example 1: Finding the Square of a Matrix
For write as a multiple of .
Before attempting to write as a multiple of , we need to calculate itself. Completing the necessary matrix multiplication gives
The output matrix is the same as the original matrix , except every entry has been multiplied by . We hence find that can be written in terms of itself by the expression .
Having seen a simple example of taking the power of a matrix, we note that we will often have to deal with expressions that potentially involve multiple matrices, as well as other matrix operations. Fortunately, we should have no problems dealing with such questions, as long as we apply the same principles we have just learned.
Example 2: Evaluating Matrix Expressions Involving Powers
Consider the matrices What is ?
We should begin by calculating both and in the usual way. We calculate that
We also have that
Now that we have both and , it is straightforward to calculate that
It is probably unsurprising that we can easily take, for instance, the third power of a matrix by employing our understanding of how we find the second power of a matrix, as we have done above.
Let us investigate how the third power of a matrix works. By definition, the third power of a square matrix is given by
Note that using the associative property of matrix multiplication, along with the definition of , we can write the right-hand side of this as
Alternatively, we can use associativity on the last two terms to write this as
So, we have shown that . In other words, once we have computed , we can find by multiplying on the right (or the left) by .
Having seen how exponentiation works for squaring and cubing, we might imagine we can apply the same principles to any power of . With the following definition, this is possible.
Definition: Power of a Matrix
If is a square matrix and is a positive integer, the power of is given by where there are copies of matrix .
In addition to this definition, we note that, using the same logic as above, it is possible to compute (for any positive integer ) by computing first and multiplying by an additional on the right or left. So, for instance, , and so on.
Let us now consider an example where we have to compute the third power of a matrix.
Example 3: Calculating Higher Powers of Matrices
Given the matrix calculate .
We should begin by calculating and then using this result to calculate . We find that
Now, we have both of the matrices which means that we can calculate as the matrix multiplication between and :
We now have everything necessary to calculate the required expression:
Up until now, we have only seen calculations involving matrices, but the extension to higher orders of square matrices is very natural. Let us now see an example of how we would find the power of a matrix.
Example 4: Squaring a 3 × 3 Matrix
The matrix has order , which means that will also have this order. Therefore, we expect to find a matrix of the form where the entries are to be calculated. We will complete the matrix multiplication in full, illustrating every step completely.
First, we calculate the entry in the first row and first column of the rightmost matrix:
The calculation is . Now, we calculate the entry in the first row and second column of the rightmost matrix:
The calculation is . Next, we focus on the entry in the first row and third column of the rightmost matrix:
The calculation is . Now, we move onto the second row of the rightmost matrix, resetting to the first column:
The calculation is . Then, we take the entry in the second row and second column:
The calculation is . The final entry in the second row is then computed:
The calculation is . The entry in the third row and first column is calculated:
The calculation is . The penultimate entry is then completed:
The calculation is . The final entry is then worked out:
The calculation is . Now that all entries of the rightmost matrix have been found, we can write the answer as
Given that taking the power of a matrix involves repeating matrix multiplication, we could reasonably expect that the algebraic rules of matrix multiplication would, to some extent, influence the rules of matrix exponentiation in a similar way. Even though this is obvious to an extent, it is dangerous to turn to the rules of conventional algebra when completing questions involving matrices under the assumption that they will still hold. In the following example, we will treat each statement individually and will present the relevant properties of matrix multiplication in tandem, explaining why the given statements do or do not hold as a result.
Example 5: Verifying Properties of Powers of Matrices
Which of the following statements is true for all matrices and ?
- Matrix multiplication is associative, which means that . We could continue this role to obtain results such as , and so forth. In the given equation, the left-hand side is , which by definition can be written as . Given the associativity property of matrix multiplication, we can write that and hence confirm that the given statement is true.
- Conventional algebra is commutative over multiplication. For two real numbers and , this means that . This result allows us to take an expression such as and use the commutative property to collect the two middle terms of the right-hand side: However, matrix multiplication is generally not commutative, meaning that except in special circumstances (such as diagonal matrices or simultaneously diagonal matrices). Therefore, the expansion cannot be simplified under the assumption that . Hence, the given statement is false.
- To complete the matrix multiplication , we can begin by writing where we have used the associativity property to arrange the final expression. Because matrix multiplication is not commutative, the bracketed term cannot be rearranged as , meaning that we cannot rewrite the final expression as , which would have allowed the simplification . Given that this is not the case, the statement is false.
- We have that Since it is generally the case that , we cannot obtain the simplification given in the question.
- We begin by completing the expansion We know that, generally, , which means that we cannot write the right-hand side as and hence the statement in the question is false.
Therefore, the correct answer is option A.
Despite the fact that some conventional rules of algebra do not hold for matrices, there are still some rules that govern powers of matrices that we can rely on. In particular, the laws of exponents for numbers can be extended to matrices in the following way.
Property: Addition and Multiplication of Powers of a Matrix
If is a square matrix and and are positive integers, then
In the final example, we will consider taking a matrix to a much higher power and see how the above properties can be used in tangent with identifying a pattern in how the matrix behaves under exponentiation.
Example 6: Finding the Higher Order Power of a Matrix by Investigating the Pattern of its Powers
Fill in the blank: If , then .
As (fifty times), clearly we should avoid trying to compute it directly. Instead, let us investigate the effect that taking powers of has for small powers of and see whether we can determine a pattern.
If we multiply by itself, in other words, if we find , we have
We note that, as this is a diagonal matrix, this might be a useful form for the matrix to be in. Continuing onward, if we calculate , we have
Interestingly, the matrix is no longer diagonal. To continue investigating the pattern, let us calculate . This is
At this point, it is possible to recognize a pattern. For the even powers of , we hypothesize that the matrix is diagonal and the nonzero entries are , where is the power of the matrix. For the odd powers, this is not the case, since there is a nonzero entry in the lower-left corner and the bottom-right entry becomes negative. However, since we only need to find where 50 is an even power, we only need to consider the first case.
Let us now show how we can find using an even power of the matrix, . Recall that
We note that the scalar can be taken outside the matrix, rewriting it in the form:
This is the identity matrix times a constant. Now, we know that the identity matrix has the property where is any matrix. In particular, if , we have
We can extend this to any power of , that is
We can use this property to calculate . Let us also recall the property , which allows us to rewrite as follows:
Since we have , this means
There are many related topics that bolster the justification for studying matrix exponentiation. When working with a square matrix, it is clear that repeatedly multiplying such a matrix by itself will generally lead to results that are successively more complicated to calculate given the large numbers involved, as we have seen in several of the examples above. It is therefore advantageous to be able to reduce the complexity of these calculations as much as possible. Under certain circumstances, it is possible to diagonalize a matrix, which significantly reduces the complexity of calculating its integer powers.
Let us finish by considering the main things we have learned in this explainer.
- For a square matrix and positive integer , we define the power of a matrix by repeating matrix multiplication; for example, where there are copies of matrix on the right-hand side.
- It is important to recognize that the power of a matrix is only well defined if the matrix is a square matrix. Furthermore, if is of order , then this will be the case for , , and so on.
- Higher powers of a matrix can be calculated with reference to the lower powers of a matrix. In other words, , , and so forth.
- If is a square matrix and and are positive integers, then