What Are Commuting Matrices in Linear Algebra?
Learn what commuting matrices are in linear algebra, with definitions, examples, and key properties. Understand when matrix multiplication is commutative and its implications.
Alice Brooks
Dec 23, 2024 13 mins read
Elementary algebra introduces us to an essential concept called the Commutative Property that governs both addition and multiplication operations equally. When considering any two numbers, a and b, equations like a + b = b + a or a x b = b x a always hold true for addition or multiplication operations, respectively. This simplifies calculations while serving as the cornerstone for more intricate algebraic operations; when we enter linear algebra, however, things become more intricate; while matrix addition may adhere to this commutative property, matrix multiplication may not, thus prompting us into asking, "What are the operational rules that matrices in linear algebra follow?" and "What are commuting matrices?"
Basic Definition of Commuting Matrices
Definition of Commuting Matrices
Linear algebra defines two matrices, A and B, as commutative when their product does not depend on which order of multiplication was applied; that is, AB = BA. This property is particularly significant as matrix multiplication often isn't commutative compared with simple addition or subtraction; thus for proper commutative status to exist, they must meet this condition for them to commute properly, which has far-reaching effects across mathematical and physical applications.
Examples:
Consider the following 2 × 2 matrices:
\[ A = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} 2 & 0 \\ 1 & 2 \end{pmatrix} \]
To check if A and B commute, we compute AB and BA:
\[ AB = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 2 & 0 \\ 1 & 2 \end{pmatrix} = \begin{pmatrix} 4 & 4 \\ 1 & 2 \end{pmatrix} \]
\[ BA = \begin{pmatrix} 2 & 0 \\ 1 & 2 \end{pmatrix} \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 2 & 4 \\ 1 & 4 \end{pmatrix} \]
Since AB ≠ BA, these matrices do not commute. Now, consider two diagonal matrices:
\[ C = \begin{pmatrix} 3 & 0 \\ 0 & 4 \end{pmatrix}, \quad D = \begin{pmatrix} 5 & 0 \\ 0 & 6 \end{pmatrix} \]
\[ CD = \begin{pmatrix} 3 & 0 \\ 0 & 4 \end{pmatrix} \begin{pmatrix} 5 & 0 \\ 0 & 6 \end{pmatrix} = \begin{pmatrix} 15 & 0 \\ 0 & 24 \end{pmatrix} \]
\[ DC = \begin{pmatrix} 5 & 0 \\ 0 & 6 \end{pmatrix} \begin{pmatrix} 3 & 0 \\ 0 & 4 \end{pmatrix} = \begin{pmatrix} 15 & 0 \\ 0 & 24 \end{pmatrix} \]
Here, CD = DC, so C and D commute.
Key Rules for Matrix Operations in Linear Algebra
Matrix calculations in linear algebra do not follow exactly the same operational rules as elementary algebra: Addition has the commutative property, while multiplication does not.
Matrix addition in linear algebra follows the commutative property, such that for any pair of identical-dimension matrices A and B with equal dimensions, A + B = B + A. This typically holds when it comes to matrix multiplication between A and B, as this requires adding products from both rows and columns of one matrix against those of another matrix; its order of multiplication becomes important here.
Examples:
To illustrate, consider the matrices:
\[ A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \]
For addition:
\[ A + B = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} + \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 3 \\ 4 & 4 \end{pmatrix} \]
\[ B + A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} + \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} = \begin{pmatrix} 1 & 3 \\ 4 & 4 \end{pmatrix} \]
Clearly, A + B = B + A.
For multiplication:
\[ AB = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 2 & 1 \\ 4 & 3 \end{pmatrix} \]
\[ BA = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} = \begin{pmatrix} 3 & 4 \\ 1 & 2 \end{pmatrix} \]
Here, AB ≠ BA, demonstrating that matrix multiplication is not commutative.
Characteristics of Commuting Matrices
Commutative Matrices Preserve Each Other's Eigenspaces
One notable characteristic of commuting matrices is their ability to maintain each other's eigenspaces. If A and B are both commuting matrices with vector v having eigenvalue l, this vector must also appear in B's eigenvector list, or else it would significantly change its physical structure if action taken against A's eigenspace by B is not significantly affecting it physically. This property ensures that changes taken against its physical structure by either party do not significantly change A.
As an illustration, let us consider two commuting matrices A and B that commute, such that \(Av = \lambda v\) for some eigenvector v and eigenvalue l; since both A and B commute, we obtain:
\[ ABv = BAv = B(\lambda v) = \lambda Bv \]
This equation shows that Bv is also an eigenvector of A corresponding to the same eigenvalue λ. Therefore, the eigenspace associated with λ for A is invariant under the action of B.
Two Commutative Matrices Share a Common Set of Eigenvectors
Commuting matrices have the distinct advantage of sharing common sets of eigenvectors, creating an integral basis of vector space consisting of simultaneously-eigenvectored bases for both matrices. This feature makes linear transformation analysis simpler.
Consider two commuting matrices A and B. If v is an eigenvector of A with eigenvalue \(\lambda_A\) and also an eigenvector of B with eigenvalue \(\lambda_B\), then:
\[ Av = \lambda_A v \]
\[ Bv = \lambda_B v \]
Given A and B commute, we can use their proximity to discover a shared set of eigenvectors, which will enable diagonalizing both matrices simultaneously—thus simplifying many problems associated with linear algebra and its applications.
The Non-Transitivity of Commutative Matrices
Commuting matrices have an inherent feature known as nontransitivity that distinguishes them from conventional lists: even when A commutes with B, and B commutes with C, this doesn't automatically imply that A also commutes with C—something to bear in mind when working with these structures. Although initially confusing, keeping this principle in mind can prove invaluable when working with these structures.
Consider, for example, matrices:
\[ A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad C = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} \]
Here, A and B commute, and B and C commute, but A and C do not commute.
\[ AB = BA = A \]
\[ BC = CB = C \]
\[ AC = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix} \]
\[ CA = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 1 & 2 \end{pmatrix} \]
Since AC ≠ CA, A and C do not commute, illustrating the non-transitivity of commuting matrices.
Theorems Related to Commuting Matrices
Theorem 1: Any Two Diagonal Matrices Are Commutative Matrices
One fundamental theorem in linear algebra states that any two diagonal matrices are commutative; that is, their product will always equal each other regardless of which order multiplication takes place—in other words, AB = BA.
Proof:
Consider two n × n diagonal matrices A and B:
\[ A = \begin{pmatrix} a_1 & 0 & \cdots & 0 \\ 0 & a_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & a_n \end{pmatrix}, \quad B = \begin{pmatrix} b_1 & 0 & \cdots & 0 \\ 0 & b_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & b_n \end{pmatrix} \]
The product AB is given by:
\[ AB = \begin{pmatrix} a_1 & 0 & \cdots & 0 \\ 0 & a_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & a_n \end{pmatrix} \begin{pmatrix} b_1 & 0 & \cdots & 0 \\ 0 & b_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & b_n \end{pmatrix} = \begin{pmatrix} a_1 b_1 & 0 & \cdots & 0 \\ 0 & a_2 b_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & a_n b_n \end{pmatrix} \]
Similarly, the product BA is:
\[ BA = \begin{pmatrix} b_1 & 0 & \cdots & 0 \\ 0 & b_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & b_n \end{pmatrix} \begin{pmatrix} a_1 & 0 & \cdots & 0 \\ 0 & a_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & a_n \end{pmatrix} = \begin{pmatrix} b_1 a_1 & 0 & \cdots & 0 \\ 0 & b_2 a_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & b_n a_n \end{pmatrix} \]
Since scalar multiplication is commutative, \(a_i b_i = b_i a_i\) for all i. Therefore, AB = BA, proving that any two diagonal matrices are commutative.
Theorem 2: Different Powers of the Same Matrix Are Commutative
Another important theorem states that different powers of the same matrix are commutative. This means that for a given matrix A, any two powers \(A^m\) and \(A^n\) commute, i.e., \(A^m A^n = A^n A^m\).
Proof:
Let A be an n × n matrix. We need to show that \(A^m\) and \(A^n\) commute for any non-negative integers m and n.
Consider the products \(A^m A^n\) and \(A^n A^m\):
\[ A^m A^n = A^{m+n} \]
\[ A^n A^m = A^{n+m} \]
Since matrix multiplication is associative, the order in which we multiply the matrices does not affect the result. Therefore, \(A^{m+n} = A^{n+m}\), which implies that:
\[ A^m A^n = A^n A^m \]
This proves that different powers of the same matrix are commutative.
Theorem 3: The Binomial Theorem for Commutative Matrices
The binomial theorem, which is well-known for scalars, also applies to commutative matrices. If A and B are two commuting matrices, then:
\[ (A + B)^n = \sum_{k=0}^{n} \binom{n}{k} A^k B^{n-k} \]
Proof:
Let A and B be two commuting matrices, i.e., AB = BA. We will use mathematical induction to prove the binomial theorem for these matrices.
Base case: For n = 1,
\[ (A + B)^1 = A + B \]which trivially satisfies the binomial theorem.
Inductive step: Assume that the binomial theorem holds for some integer n:
\[ (A + B)^n = \sum_{k=0}^{n} \binom{n}{k} A^k B^{n-k} \]
We need to show that it holds for n + 1:
\[ (A + B)^{n+1} = (A + B)(A + B)^n \]
Using the inductive hypothesis,
\[ (A + B)^{n+1} = (A + B) \sum_{k=0}^{n} \binom{n}{k} A^k B^{n-k} \]
Since A and B commute, we can distribute A + B through the sum:
\[ (A + B)^{n+1} = \sum_{k=0}^{n} \binom{n}{k} (A + B) A^k B^{n-k} \]
\[ = \sum_{k=0}^{n} \binom{n}{k} (A^{k+1} B^{n-k} + A^k B^{n-k+1}) \]
Reindexing the sums and combining terms, we get:
\[ (A + B)^{n+1} = A^{n+1} + \sum_{k=1}^{n} \left( \binom{n}{k-1} + \binom{n}{k} \right) A^k B^{n+1-k} + B^{n+1} \]
Using the identity \(\binom{n}{k-1} + \binom{n}{k} = \binom{n+1}{k}\),
\[ (A + B)^{n+1} = \sum_{k=0}^{n+1} \binom{n+1}{k} A^k B^{n+1-k} \]
Thus, the binomial theorem holds for n + 1, completing the induction.
Theorem 4: If Matrices A and B Can Be Diagonalized by the Same Invertible Matrix P, Then A and B Are Commutative
This theorem states that if two matrices, A and B, can be simultaneously diagonalized by the same invertible matrix, P, then A and B commute. In other words, if there exists an invertible matrix P such that \(P^{-1}AP\) and \(P^{-1}BP\) are both diagonal matrices, then AB = BA.
Proof:
Suppose A and B can be diagonalized by the same invertible matrix P. This means there exist diagonal matrices \(D_A\) and \(D_B\) such that:
\[ P^{-1}AP = D_A \]
\[ P^{-1}BP = D_B \]
To show that A and B commute, we need to prove that AB = BA.
Consider the product AB:
\[ AB = A(PP^{-1}) B = A(PD_BP^{-1}) = (P D_A P^{-1})(P D_B P^{-1}) \]
Since \(P^{-1}P = I\), the identity matrix, we have:
\[ AB = P D_A (P^{-1} P) D_B P^{-1} = P D_A D_B P^{-1} \]
Similarly, consider the product \(BA\):
\[ BA = B(PP^{-1}) A = B(PD_A P^{-1}) = (P D_B P^{-1})(P D_A P^{-1}) \]
\[ BA = P D_B (P^{-1} P) D_A P^{-1} = P D_B D_A P^{-1} \]
Since \(D_A\) and \(D_B\) are diagonal matrices, they commute (as shown in Theorem 1). Therefore, \(D_A D_B = D_B D_A\), and we have:
\[ AB = P D_A D_B P^{-1} = P D_B D_A P^{-1} = BA \]
Other Theorems
Theorem 1:
If Two Matrices A and B Have Equal Eigenvectors, They Are Commutative
Theorem 2:
When two matrices, A and B, are commutative with their minimum polynomials matching their characteristic polynomials with maximum degree, matrix B can be expressed as polynomials of matrix A.
Theorem 3:
Given any two Hermitian matrices, A and B, with similar eigenspaces, which then commute (AB = BA being equal) and share an orthonormal basis of eigenvectors that commute, then these A/B matrix pairs also share an orthonormal basis of eigenvectors between themselves that also commute.
Historical Development of Commuting Matrices
Early Origins in the 19th Century
Commuting matrices were introduced during the 19th century through James Joseph Sylvester and Arthur Cayley—two influential mathematicians of that era. Sylvester introduced the term "matrix" in 1850 while explaining its nature as algebraic objects with properties including commutativity for simplifying systems of linear equations; Cayley made significant contributions that expanded this field further.
Arthur Cayley's 1858 paper on the Cayley-Hamilton Theorem established modern matrix theory by showing how matrices satisfied their characteristic polynomials. Additionally, Cayley and Sylvester demonstrated how commuting matrices could be diagonalized simultaneously for diagonalization purposes, further emphasizing their importance and opening new avenues in mathematics and applied sciences.
Maturation and Applications in the 20th Century
Since 1900, significant advances in commuting matrices have been achieved, particularly in quantum mechanics and linear algebra. Commutation relationships among operators play an essential part in quantum theory, such as Heisenberg's uncertainty principle. Commuting matrices can instantly diagonalize to simplify linear transformations and system analysis—an ability that finds application across stability analysis, dynamical system modeling, and numerical linear algebra, providing more efficiency and accuracy during matrix computations.
An ability to simultaneously diagonalize matrices greatly simplifies the complexity of matrix operations, leading to advantages in applications ranging from computer graphics and engineering simulations to financial transactions and statistical forecasting, statistical irrationality testing, and computer vision research. Advancements in understanding commuting matrices over the last century have had an incredible transformative effect on both theoretical and applied mathematics research, expanding upon work by 19th-century mathematicians while shaping modern science and engineering research projects.
Applications of Commuting Matrices in Real Life
Applications of Commutative Matrices in Economics
Commuting matrices play an integral part in economics, particularly in input-output models, optimization problems, and economic forecasting. Commuting matrices serve an analytical function by guaranteeing that the ordering of operations does not alter overall results in input-output models, helping economists better comprehend sector interdependencies. Leontief input-output models use Leontief matrices to represent inputs needed from one sector for producing output in another; when these commute, their computation simplifies overall economic output as well as impacts due to changes in specific sectors, assisting policymakers when planning interventions and analyzing shocks more quickly and comprehensively than before.
Commuting matrices can aid the solution process for optimization problems involving linear programming, especially those involving linear programming with constraints, in a similar fashion to economic models that include optimizing an objective function subject to linear constraints. Furthermore, their commutative properties allow more reliable forecasts, which inform decisions more precisely on monetary and fiscal policies.
Applications of Commutative Matrices in Communication Systems
Commuting matrices play an essential role in communication systems for signal processing, error correction, and network analysis. By linking various stages of signal transformation together in their transit time sequences, commuting matrices streamline protocol design and analysis while guaranteeing their sequence does not influence final output. MIMO systems utilizing multiple input multiple output systems also benefit greatly from using such optimization in creating efficient detection/decoding algorithms to enhance system performance and thus increase overall system functionality.
Error correction codes also take advantage of commuting matrices, making their processing faster and more reliable—essential features in high-speed communication systems for maintaining data integrity. Network analysis uses a matrix representation of connections and flows as a basis to optimize optimization efforts to maximize data throughput while decreasing latency; overall, commuting matrices simplify complex operations to increase efficiency and reliability within communication systems essential in today's interconnected society.
