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Little Devil
Why might two matrices commute? I.e Why would AB=BA because in general, matrices usually do not commute. What are the properties of matrices that do commute?
Ben
Ben
rick1138 said:Diagonal matrices commute.
Tola said:Can you give me and example to find matrices commute with matrices A that we know?
Matrix multiplication is an operation used to combine two matrices to produce a third matrix. It is a fundamental operation in linear algebra and is defined for matrices where the number of columns in the first matrix matches the number of rows in the second matrix.
To multiply two matrices A and B, you take the dot product of rows from matrix A with columns from matrix B. The result is a new matrix C, where each element C[i][j] is computed as the sum of the products of elements from row i of A and column j of B.
Two matrices, A and B, are said to "commute" if their product AB is equal to the product BA. In other words, if AB = BA, then A and B commute.
No, not all matrices commute. Whether matrices commute or not depends on their specific properties and elements. In general, matrix multiplication is not commutative, meaning that AB is not necessarily equal to BA for arbitrary matrices A and B.
Yes, there are special cases where matrices commute. Some matrices, particularly diagonal matrices (matrices where all off-diagonal elements are zero), commute with many other matrices. Additionally, the identity matrix (a special diagonal matrix) commutes with any matrix.
When matrices commute (AB = BA), it simplifies certain mathematical operations and computations involving those matrices. Commuting matrices often have special properties that can be useful in linear algebra and applications in physics, engineering, and other fields.
Yes, matrix commutation has applications in various fields, including quantum mechanics, quantum computing, physics, and control theory. In quantum mechanics, for example, operators representing physical observables often need to commute to have simultaneous eigenstates, which is crucial in quantum physics calculations.
You can learn more about matrix multiplication and commutation by studying linear algebra textbooks, taking courses in linear algebra and matrix theory, and exploring online resources and tutorials dedicated to these topics. Practical exercises and problem-solving can enhance your understanding.