Fiche explicative de la leçon: Power of a Matrix | Nagwa Fiche explicative de la leçon: Power of a Matrix | Nagwa

Fiche explicative de la leçon: Power of a Matrix Mathématiques

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 𝑥 and raise it to a power 𝑦, giving 𝑥. With the exception of taking zero to a negative power, it does not matter whether 𝑥 or 𝑦 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 𝐴 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 𝐴 is a square matrix, 𝐴 is defined by 𝐴=𝐴×𝐴.

In other words, just like for the exponentiation of numbers (i.e., 𝑎=𝑎×𝑎), 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 𝐴 must be square. This is because, for two general matrices 𝐴 and 𝐵, the matrix multiplication 𝐴𝐵 is only well defined if there is the same number of columns in 𝐴 as there are rows in 𝐵. If 𝐴 has order 𝑚×𝑛 and 𝐵 has order 𝑛×𝑝, then 𝐴𝐵 is well defined and has order 𝑚×𝑛. If we were only to consider the matrix 𝐴 and attempt to complete the matrix multiplication 𝐴=𝐴×𝐴, then we would be attempting to multiply a matrix with order 𝑚×𝑛 by another matrix with order 𝑚×𝑛. This can only be well defined if 𝑚=𝑛, meaning that 𝐴 has to be a matrix with order 𝑛×𝑛 (in other words square). The order of 𝐴 is therefore identical to the original matrix 𝐴.

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 𝐴 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 𝐴=1325.

To calculate matrix 𝐴, we are multiplying the matrix 𝐴 by itself. In other words, we have 𝐴=𝐴×𝐴=13251325.

As expected, this multiplication is well defined, since we have a 2×2 matrix multiplied by a 2×2 matrix. It now remains to complete the matrix multiplication, which we can do for each entry (𝑖,𝑗) by multiplying the elements in row 𝑖 of the left matrix by the elements in column 𝑗 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 𝐴=5181219.

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 𝐴=4545, write 𝐴 as a multiple of 𝐴.

Answer

Before attempting to write 𝐴 as a multiple of 𝐴, we need to calculate 𝐴 itself. Completing the necessary matrix multiplication gives 𝐴=𝐴×𝐴=45454545=4545.

The output matrix 𝐴 is the same as the original matrix 𝐴, except every entry has been multiplied by 1. We hence find that 𝐴 can be written in terms of itself by the expression 𝐴=𝐴.

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 𝑋=3356,𝑌=1366. What is 𝑋𝑌?

Answer

We should begin by calculating both 𝑋 and 𝑌 in the usual way. We calculate that 𝑋=𝑋×𝑋=33563356=6274521.

We also have that 𝑌=𝑌×𝑌=13661366=19153054.

Now that we have both 𝑋 and 𝑌, it is straightforward to calculate that 𝑋𝑌=627452119153054=25421533.

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 𝐴 is given by 𝐴=𝐴×𝐴×𝐴.

Note that using the associative property of matrix multiplication, along with the definition of 𝐴, we can write the right-hand side of this as 𝐴×𝐴×𝐴=(𝐴×𝐴)×𝐴=𝐴×𝐴.

Alternatively, we can use associativity on the last two terms to write this as 𝐴×𝐴×𝐴=𝐴×(𝐴×𝐴)=𝐴×𝐴.

So, we have shown that 𝐴=𝐴𝐴=𝐴𝐴. In other words, once we have computed 𝐴, we can find 𝐴 by multiplying 𝐴 on the right (or the left) by 𝐴.

Having seen how exponentiation works for squaring and cubing, we might imagine we can apply the same principles to any power of 𝐴. With the following definition, this is possible.

Definition: Power of a Matrix

If 𝐴 is a square matrix and 𝑘 is a positive integer, the 𝑘th power of 𝐴 is given by 𝐴=𝐴×𝐴××𝐴, where there are 𝑘 copies of matrix 𝐴.

In addition to this definition, we note that, using the same logic as above, it is possible to compute 𝐴 (for any positive integer 𝑘) by computing 𝐴 first and multiplying by an additional 𝐴 on the right or left. So, for instance, 𝐴=𝐴×𝐴=𝐴×𝐴, 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 𝐴=4037, calculate 𝐴3𝐴.

Answer

We should begin by calculating 𝐴 and then using this result to calculate 𝐴. We find that 𝐴=𝐴×𝐴=40374037=1603349.

Now, we have both of the matrices 𝐴=4037,𝐴=1603349, which means that we can calculate 𝐴 as the matrix multiplication between 𝐴 and 𝐴: 𝐴=𝐴×𝐴=40371603349=640279343.

We now have everything necessary to calculate the required expression: 𝐴3𝐴=64027934331603349=64027934348099147=160180196.

Up until now, we have only seen calculations involving 2×2 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 3×3 matrix.

Example 4: Squaring a 3 × 3 Matrix

Consider 𝐴=112101210.

Find 𝐴.

Answer

The matrix 𝐴 has order 3×3, which means that 𝐴 will also have this order. Therefore, we expect to find a matrix of the form 𝐴=𝐴×𝐴=112101210112101210=, 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: 112101210112101210=6.

The calculation is 1×1+1×1+2×2=6. Now, we calculate the entry in the first row and second column of the rightmost matrix: 112101210112101210=63.

The calculation is 1×1+1×0+2×1=3. Next, we focus on the entry in the first row and third column of the rightmost matrix: 112101210112101210=633.

The calculation is 1×2+1×1+2×0=3. Now, we move onto the second row of the rightmost matrix, resetting to the first column: 112101210112101210=6333.

The calculation is 1×1+0×1+1×2=3. Then, we take the entry in the second row and second column: 112101210112101210=63332,

The calculation is 1×1+0×0+1×1=2. The final entry in the second row is then computed: 112101210112101210=633322.

The calculation is 1×2+0×1+1×0=2. The entry in the third row and first column is calculated: 112101210112101210=6333223.

The calculation is 2×1+1×1+0×2=3. The penultimate entry is then completed: 112101210112101210=63332232.

The calculation is 2×1+1×0+0×1=2. The final entry is then worked out: 112101210112101210=633322325.

The calculation is 2×2+1×1+0×0=5. Now that all entries of the rightmost matrix have been found, we can write the answer as 𝐴=633322325.

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 𝑛×𝑛 matrices 𝐴 and 𝐵?

  1. 𝐴𝐵=𝐴(𝐴𝐵)𝐵
  2. (𝐴𝐵)=𝐴2𝐴𝐵+𝐵
  3. (𝐴𝐵)=𝐴𝐵
  4. (𝐴+𝐵)=𝐴+2𝐴𝐵+𝐵
  5. (𝐴+𝐵)(𝐴𝐵)=𝐴𝐵

Answer

  1. Matrix multiplication is associative, which means that 𝐴(𝐵𝐶)=(𝐴𝐵)𝐶. We could continue this role to obtain results such as (𝐴𝐵)(𝐶𝐷)=𝐴(𝐵𝐶)𝐷=𝐴𝐵𝐶𝐷, and so forth. In the given equation, the left-hand side is 𝐴𝐵, which by definition can be written as 𝐴𝐵=𝐴𝐴𝐵𝐵. Given the associativity property of matrix multiplication, we can write that 𝐴𝐵=𝐴(𝐴𝐵)𝐵 and hence confirm that the given statement is true.
  2. Conventional algebra is commutative over multiplication. For two real numbers 𝑎 and 𝑏, this means that 𝑎𝑏=𝑏𝑎. This result allows us to take an expression such as (𝑎𝑏)=𝑎𝑎𝑏𝑏𝑎+𝑏 and use the commutative property to collect the two middle terms of the right-hand side: (𝑎𝑏)=𝑎2𝑎𝑏+𝑏. However, matrix multiplication is generally not commutative, meaning that 𝐴𝐵𝐵𝐴 except in special circumstances (such as diagonal matrices or simultaneously diagonal matrices). Therefore, the expansion (𝐴𝐵)=𝐴𝐴𝐵𝐵𝐴+𝐵 cannot be simplified under the assumption that 𝐴𝐵=𝐵𝐴. Hence, the given statement is false.
  3. To complete the matrix multiplication (𝐴𝐵), we can begin by writing (𝐴𝐵)=(𝐴𝐵)(𝐴𝐵)=𝐴(𝐵𝐴)𝐵, where we have used the associativity property to arrange the final expression. Because matrix multiplication is not commutative, the bracketed term (𝐵𝐴) cannot be rearranged as (𝐴𝐵), meaning that we cannot rewrite the final expression as 𝐴𝐴𝐵𝐵, which would have allowed the simplification 𝐴𝐵. Given that this is not the case, the statement is false.
  4. We have that (𝐴+𝐵)=𝐴+𝐴𝐵+𝐵𝐴+𝐵. Since it is generally the case that 𝐴𝐵𝐵𝐴, we cannot obtain the simplification given in the question.
  5. We begin by completing the expansion (𝐴+𝐵)(𝐴𝐵)=𝐴+𝐵𝐴𝐴𝐵𝐵. We know that, generally, 𝐵𝐴𝐴𝐵, which means that we cannot write the right-hand side as 𝐴𝐵 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 𝐴 is a square matrix and 𝑟 and 𝑠 are positive integers, then 𝐴𝐴=𝐴,(𝐴)=𝐴.

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 𝐴=4034, then 𝐴=.

Answer

As 𝐴=𝐴×𝐴××𝐴 (fifty times), clearly we should avoid trying to compute it directly. Instead, let us investigate the effect that taking powers of 𝐴 has for small powers of 𝐴 and see whether we can determine a pattern.

If we multiply 𝐴 by itself, in other words, if we find 𝐴=𝐴×𝐴, we have 𝐴=40344034=4004.

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 𝐴=𝐴×𝐴, we have 𝐴=40044034=40434.

Interestingly, the matrix is no longer diagonal. To continue investigating the pattern, let us calculate 𝐴=𝐴×𝐴. This is 𝐴=404344034=4004.

At this point, it is possible to recognize a pattern. For the even powers of 𝐴, we hypothesize that the matrix is diagonal and the nonzero entries are 4, where 𝑛 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 𝐴 where 50 is an even power, we only need to consider the first case.

Let us now show how we can find 𝐴 using an even power of the matrix, 𝐴. Recall that 𝐴=4004.

We note that the scalar 4 can be taken outside the matrix, rewriting it in the form: 𝐴=41001.

This is the 2×2 identity matrix 𝐼 times a constant. Now, we know that the identity matrix has the property 𝐼𝑋=𝑋𝐼=𝑋, where 𝑋 is any 2×2 matrix. In particular, if 𝑋=𝐼, we have 𝐼=𝐼×𝐼=𝐼.

We can extend this to any power of 𝐼, that is 𝐼=𝐼.

We can use this property to calculate 𝐴. Let us also recall the property (𝐴)=𝐴, which allows us to rewrite 𝐴 as follows: 𝐴=𝐴.

Since we have 𝐴=4𝐼, this means 𝐴=4𝐼=4𝐼=4𝐼=21001.

Since, 4=2.

Then, 4=2=2.

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.

Key Points

  • For a square matrix 𝐴 and positive integer 𝑘, we define the power of a matrix by repeating matrix multiplication; for example, 𝐴=𝐴×𝐴××𝐴, where there are 𝑘 copies of matrix 𝐴 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 𝐴 is of order 𝑛×𝑛, then this will be the case for 𝐴, 𝐴, and so on.
  • Higher powers of a matrix can be calculated with reference to the lower powers of a matrix. In other words, 𝐴=𝐴×𝐴, 𝐴=𝐴×𝐴, and so forth.
  • If 𝐴 is a square matrix and 𝑟 and 𝑠 are positive integers, then 𝐴𝐴=𝐴,(𝐴)=𝐴.

Télécharger l’application Nagwa Classes

Assistez à des séances, chattez avec votre enseignant ainsi que votre classe et accédez à des questions spécifiques à la classe. Téléchargez l’application Nagwa Classes dès aujourd’hui !

Nagwa utilise des cookies pour vous garantir la meilleure expérience sur notre site web. Apprenez-en plus à propos de notre Politique de confidentialité