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netheril96
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I googled for a proof,but didn't find one.
Could anyone give me a link to a proof?
Could anyone give me a link to a proof?
Mark44 said:Is this a homework or test problem? If so, you're not likely to get much help without showing what you have tried.
Commuting matrices are matrices that can be multiplied in any order without changing the result. In other words, if we have two matrices A and B, they are considered commuting matrices if AB = BA.
When two matrices commute, it means that they share a common set of eigenvectors. This is important because it allows us to simplify calculations and make it easier to find the eigenvalues and eigenvectors of the matrices.
Yes, commuting matrices can have different eigenvalues. The important factor is that they have the same eigenvectors. This means that they may have different scaling factors, but the directions of the eigenvectors are the same.
Commuting matrices do not change the result of matrix multiplication, as they can be multiplied in any order. This is because the order of multiplication does not affect the shared eigenvectors. However, it is important to note that commuting matrices do not always exist for all matrices.
Commuting matrices have various applications in fields such as physics, engineering, and computer science. For example, in quantum mechanics, commuting matrices represent observables that can be measured simultaneously. In computer science, they are used in algorithms for matrix multiplication, which are essential for data analysis and machine learning.