Explainer: Power of a Matrix

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 operations in linear algebra that are very similar to the well-known operations from conventional algebra, such as addition, subtraction, and scaling. There are also additional operations such as matrix multiplication and inversion which to some extent mirror the algebraic properties of their conventional counterparts, while being calculated in a way that is fundamentally more complex.

One operation that is central to both conventional algebra and linear algebra is that of exponentiation, which is usually referred to as taking the β€œpower” of a number or matrix. In conventional algebra, it is possible to take any number π‘₯ and raise it to a power 𝑦, giving π‘₯. It does not matter whether π‘₯ and 𝑦 are zero, nonzero, integer, noninteger, rational, irrational, or complex as the output can always be calculated. The same is not true in linear algebra, where a matrix 𝐴 cannot always be exponentiated. For example, as we will shortly see, it is not possible to exponentiate a nonsquare matrix because the operation will not be well defined. As another example, we cannot raise 𝐴 to the power βˆ’1 or any other negative power unless the matrix inverse 𝐴 is known to exist, which is only possible for square matrices with a nonzero determinant. These are just two of the exceptions for dealing with exponentiation in linear algebra, which we now define more rigorously.

Definition: Power of a Square Matrix

For a square matrix 𝐴 and positive integer π‘˜, the π‘˜th power of 𝐴 is defined by multiplying this matrix by itself repeatedly; that is, 𝐴=𝐴×𝐴×⋯×𝐴, where there are π‘˜ copies of the matrix 𝐴.

It is easiest to demonstrate this definition with a simple, nontrivial example. We define the matrix 𝐴=1βˆ’325.

To calculate the matrix 𝐴, we are multiplying the matrix 𝐴 by itself. In other words, we could write 𝐴=𝐴×𝐴=1βˆ’3251βˆ’325.

It now remains to complete the matrix multiplication. If a matrix 𝑀 has order π‘šΓ—π‘› and another 𝑁 has order 𝑛×𝑝, then the matrix multiplication 𝑀𝑁 is well defined, resulting in a matrix of order π‘šΓ—π‘. In the equation above, we are multiplying a matrix of order 2Γ—2 by a matrix of order 2Γ—2, meaning that the output matrix will also have order 2Γ—2. We must therefore find the matrix on the right-hand side of the equation 1βˆ’3251βˆ’325=ο’βˆ—βˆ—βˆ—βˆ—οž, where the βˆ— entries are to be found. We complete the calculations with reference to the definition of matrix multiplication. First, we consider the first row of the leftmost matrix and the first column of the middle matrix, which corresponds to the entry in the first row and first column of the rightmost matrix, as follows: 1βˆ’3251βˆ’325=ο“βˆ’5βˆ—βˆ—βˆ—οŸ, where we have completed the calculation 1Γ—1+(βˆ’3)Γ—2=βˆ’5. Next, we take the entry in the first row of the leftmost matrix and the second column of the middle matrix, which we will combine to find the entry in the first row and second column of the rightmost matrix: 1βˆ’3251βˆ’325=ο“βˆ’5βˆ’18βˆ—βˆ—οŸ, where we have calculated 1Γ—(βˆ’3)+(βˆ’3)Γ—5=βˆ’18. Next, we use the second row of the leftmost matrix and the first column of the middle matrix: 1βˆ’3251βˆ’325=ο”βˆ’5βˆ’1812βˆ—ο , with the calculations being 2Γ—1+5Γ—2=12. The entry in the second row and second column of the rightmost matrix is then calculated: 1βˆ’3251βˆ’325=ο”βˆ’5βˆ’181219, where we have calculated that 2Γ—(βˆ’3)+5Γ—5=19. Now that all entries have been computed, we can write that 𝐴=ο”βˆ’5βˆ’181219.

In this specific example, we have demonstrated how to take the square of a matrix. Naturally, the definition can be extended to higher powers. Given that we have calculated both 𝐴 and 𝐴, we could also calculate 𝐴. First, we write the two matrices 𝐴=1βˆ’325,𝐴=ο”βˆ’5βˆ’181219.

Then, completing the matrix multiplication directly gives 𝐴=𝐴×𝐴=1βˆ’325ο ο”βˆ’5βˆ’181219=ο”βˆ’41βˆ’755059.

Likewise, we could calculate 𝐴οŠͺ, 𝐴, and so on. Before practicing some further examples of taking the power of a matrix, we will provide one further result that will explain why we may only take the power of a square matrix.

Theorem: Matrix Powers and Square Matrices

Taking the power of a matrix 𝐴 is only well defined if this is a square matrix. If 𝐴 has order 𝑛×𝑛, then this order will be common to 𝐴, 𝐴, 𝐴οŠͺ, and so on.

For two matrices 𝐴 and 𝐡, the matrix multiplication 𝐴𝐡 is only well defined if there is the same number of columns in 𝐴 as there are rows in 𝐡. If 𝐴 has order π‘šΓ—π‘› and 𝐡 has order 𝑛×𝑝, then 𝐴𝐡 is well defined and has order π‘šΓ—π‘. If we were only to consider the matrix 𝐴 and attempt to complete the matrix multiplication 𝐴=𝐴𝐴, then we would be attempting to multiply a matrix with order π‘šΓ—π‘› by a matrix with order π‘šΓ—π‘›. This can only be well defined if π‘š=𝑛, meaning that 𝐴 has to be a matrix with order 𝑛×𝑛, which implies that this is a square matrix. The order of 𝐴 is therefore identical to the original matrix 𝐴, as is also the case for 𝐴, 𝐴οŠͺ, and so forth.

Example 1: Finding the Square of 𝐴 Matrix

For 𝐴=4βˆ’54βˆ’5, 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=4βˆ’54βˆ’54βˆ’54βˆ’5=ο”βˆ’45βˆ’45.

The output matrix 𝐴 is the same as the original matrix 𝐴, except every entry has been multiplied by βˆ’1. We hence find that 𝐴 can be written in terms of itself by the expression 𝐴=βˆ’π΄οŠ¨.

It is seldom the case that we wish to consider the power of a matrix in isolation and, normally, we would be combining any matrix powers with other expressions involving, potentially, other matrices. The principles of matrix exponentiation never change, so even if multiple matrices are involved then the working should never be much more difficult.

Example 2: Evaluating Matrix Expressions Involving Powers

Consider the matrices 𝑋=ο”βˆ’3βˆ’35βˆ’6,π‘Œ=136βˆ’6. What is π‘‹βˆ’π‘ŒοŠ¨οŠ¨?

Answer

We should begin by calculating both π‘‹οŠ¨ and π‘ŒοŠ¨ in the usual way. We calculate that 𝑋=𝑋×𝑋=ο”βˆ’3βˆ’35βˆ’6ο ο”βˆ’3βˆ’35βˆ’6=ο”βˆ’627βˆ’4521.

We also have that π‘Œ=π‘ŒΓ—π‘Œ=136βˆ’6136βˆ’6=19βˆ’15βˆ’3054.

Now that we have both π‘‹οŠ¨ and π‘ŒοŠ¨, it is straightforward to calculate that π‘‹βˆ’π‘Œ=ο”βˆ’627βˆ’4521ο βˆ’ο”19βˆ’15βˆ’3054=ο”βˆ’2542βˆ’15βˆ’33.

It is probably unsurprising that we can easily take, for instance, the third and fourth powers of a matrix employing our understanding of how we find the second power of a matrix, as we have done above. Following the usual laws of powers and indices, for any integer π‘˜ we can write the matrix 𝐴=π΄Γ—π΄ο‡ο‰οŠ, providing that π‘š+𝑛=π‘˜. We would expect, therefore, that we can write 𝐴=π΄Γ—π΄οŠ©οŠ¨ or 𝐴=π΄Γ—π΄οŠ©οŠ¨, with the two results being the same. This is a perfectly reasonable assumption and is true, being possible to demonstrate after it has been shown that matrix multiplication is associative.

Example 3: Calculating Higher Powers of Matrices

Given the matrix 𝐴=40βˆ’37, calculate π΄βˆ’3𝐴.

Answer

We should begin by calculating 𝐴 and then using this result to calculate 𝐴. We find that 𝐴=𝐴×𝐴=40βˆ’3740βˆ’37=160βˆ’3349.

Now we have both of the matrices 𝐴=40βˆ’37,𝐴=160βˆ’3349, which means that we can calculate 𝐴 as the matrix multiplication between 𝐴 and 𝐴: 𝐴=𝐴×𝐴=40βˆ’37160βˆ’3349=640βˆ’279343.

We now have everything necessary to calculate the required expression: π΄βˆ’3𝐴=640βˆ’279343ο βˆ’3160βˆ’3349=640βˆ’279343ο βˆ’ο”480βˆ’99147=160βˆ’180196.

There are alternatives for calculating powers of a square matrix which allow for the process to be simplified. For example, it is possible to use the characteristic polynomial of a square matrix to then employ the Cayley-Hamilton theorem. For an 𝑛×𝑛 matrix 𝐴, this theorem allows for 𝐴 to be written in terms of the lower-order expressions 𝐴,𝐴,…,𝐴and and the identity matrix 𝐼. Alternatively, we could take a square matrix 𝐴 and then use the similarity equation to simplify the calculation of 𝐴 to any positive integer power. These two options are both very interesting in their own right and require in-depth study to understand, although they are both supremely versatile and elegant results. For the remainder of this explainer, we will continue with our manual method of computing the power of a matrix.

Example 4: Powers of Matrices

Consider the matrix 𝐴=112101210. Find 𝐴.

Answer

The matrix 𝐴 has order 3Γ—3, which means that 𝐴 will also have this order. Therefore, we expect to find a matrix of the form 𝐴=𝐴×𝐴=112101210112101210=ο–βˆ—βˆ—βˆ—βˆ—βˆ—βˆ—βˆ—βˆ—βˆ—ο’, where the βˆ— 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: 112101210112101210=6βˆ—βˆ—βˆ—βˆ—βˆ—βˆ—βˆ—βˆ—ο£, where the calculation we completed was 1Γ—1+1Γ—1+2Γ—2=6. Now we calculate the entry in the first row and second column of the rightmost matrix: 112101210112101210=63βˆ—βˆ—βˆ—βˆ—βˆ—βˆ—βˆ—ο£, where the calculation is 1Γ—1+1Γ—0+2Γ—1=3. Next, we focus on the entry in the first row and third column of the rightmost matrix: 112101210112101210=633βˆ—βˆ—βˆ—βˆ—βˆ—βˆ—ο£, where we have 1Γ—2+1Γ—1+2Γ—0=3. Now we move onto the second row of the rightmost matrix, resetting to the first column: 112101210112101210=6333βˆ—βˆ—βˆ—βˆ—βˆ—ο€ with the calculation being 1Γ—1+0Γ—1+1Γ—2=3. Then, we take the entry in the second row and second column: 112101210112101210=63332βˆ—βˆ—βˆ—βˆ—ο€, where 1Γ—1+0Γ—0+1Γ—1=2The final entry in the second row is then computed: 112101210112101210=633322βˆ—βˆ—βˆ—ο€, since 1Γ—2+0Γ—1+1Γ—0=2. The entry in the third row and first column is calculated: 112101210112101210=6333223βˆ—βˆ—ο€, given that 2Γ—1+1Γ—1+0Γ—2=3. The penultimate entry is then completed: 112101210112101210=63332232βˆ—ο€, where we have calculated 2Γ—1+1Γ—0+0Γ—1=2. The final entry is then worked out: 112101210112101210=633322325, given that 2Γ—2+1Γ—1+0Γ—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 matrix exponentiation in a similar way. Even though this is obvious to an extent, it is upsettingly easy 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: Properties of Raising Matrices to a Power

Which of the following statements is true for all 𝑛×𝑛 matrices 𝐴 and 𝐡?

  1. 𝐴𝐡=𝐴(𝐴𝐡)𝐡
  2. (π΄βˆ’π΅)=π΄βˆ’2𝐴𝐡+𝐡
  3. (𝐴𝐡)=𝐴𝐡
  4. (𝐴+𝐡)=𝐴+2𝐴𝐡+𝐡
  5. (𝐴+𝐡)(π΄βˆ’π΅)=π΄βˆ’π΅οŠ¨οŠ¨

Answer

  1. Matrix multiplication is associative, which means that 𝐴(𝐡𝐢)=(𝐴𝐡)𝐢. We could continue this role to obtain results such as (𝐴𝐡)(𝐢𝐷)=𝐴(𝐡𝐢)𝐷=𝐴𝐡𝐢𝐷, and so forth. In the given equation, the left-hand side is 𝐴𝐡, which by definition can be written as 𝐴𝐡=𝐴𝐴𝐡𝐡. Given the associativity property of matrix multiplication, we can write that 𝐴𝐡=𝐴(𝐴𝐡)𝐡 and hence confirm that the given statement is true.
  2. Conventional algebra is commutative over multiplication. For two real numbers π‘Ž and 𝑏, this means that π‘Žπ‘=π‘π‘Ž. This result allows us to take an expression such as (π‘Žβˆ’π‘)=π‘Žβˆ’π‘Žπ‘βˆ’π‘π‘Ž+π‘οŠ¨οŠ¨οŠ¨ and use the commutative property to collect the two middle terms of the right-hand side: (π‘Žβˆ’π‘)=π‘Žβˆ’2π‘Žπ‘+𝑏. However, matrix multiplication is generally not commutative, meaning that 𝐴𝐡≠𝐡𝐴 unless in special circumstances (such as diagonal matrices or simultaneously diagonal matrices). Therefore, the expansion (π΄βˆ’π΅)=π΄βˆ’π΄π΅βˆ’π΅π΄+𝐡 cannot be simplified under the assumption that 𝐴𝐡=𝐡𝐴. Hence, the given statement is false.
  3. To complete the matrix multiplication (𝐴𝐡), we can begin by writing (𝐴𝐡)=(𝐴𝐡)(𝐴𝐡)=𝐴(𝐡𝐴)𝐡, where we have used the associativity property to arrange the final expression. Because matrix multiplication is not commutative, the bracketed term (𝐡𝐴) cannot be rearranged as (𝐴𝐡), meaning that we cannot rewrite the final expression as 𝐴𝐴𝐡𝐡, which would have allowed the simplification 𝐴𝐡. Given that this is not the case, the statement is false.
  4. We have that (𝐴+𝐡)=𝐴+𝐴𝐡+𝐡𝐴+𝐡. Since it is generally the case that 𝐴𝐡≠𝐡𝐴, we cannot obtain the simplification given in the question.
  5. We begin by completing the expansion (𝐴+𝐡)(π΄βˆ’π΅)=𝐴+π΅π΄βˆ’π΄π΅βˆ’π΅. We know that generally 𝐡𝐴≠𝐴𝐡, meaning that we cannot write the right-hand side as π΄βˆ’π΅οŠ¨οŠ¨and hence the statement in the question is false.

There are many related topics which 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. To this end, the Cayley-Hamilton theorem provides an elegant and mathematically gratifying method for calculating the power of a matrix using the characteristic polynomial and matrix powers of a lower order. Under certain circumstances, it is possible to diagonalize a matrix, which significantly reduces the complexity of calculating its integer powers.

Key Points

  • For a square matrix 𝐴 and positive integer π‘˜, we define the power of a matrix by repeating matrix multiplication; that is, 𝐴=𝐴×𝐴×⋯×𝐴, where there are π‘˜ copies of the matrix 𝐴 on the right-hand side.
  • 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=π΄Γ—π΄οŠ©οŠ¨, 𝐴=𝐴×𝐴=𝐴×𝐴=𝐴×𝐴οŠͺ, and so forth.
  • The associative and noncommutative nature of matrix multiplication must be fully understood before attempting to simplify expressions involving the powers of a matrix.

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