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LagrangeEuler
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Is it possible to find matrices that commute but eigenvectors of one matrix are not the eigenvectors of the other one. Could you give me example for it?
LagrangeEuler said:Is it possible to find matrices that commute but eigenvectors of one matrix are not the eigenvectors of the other one. Could you give me example for it?
LagrangeEuler said:Matrix A practically do not have eigenvectors. Right? Because it is not diagonalizable.
What about two hermitian matrix. Is there any posiibility like this. Is it some easy way to construct this?
Yes. But the other one is not. Is there any example like this where both matrices ##A## and ##B## are hermitian.Math_QED said:Yes, look at the edit in my first post. (The identity matrix is hermitian)
LagrangeEuler said:Yes. But the other one is not. Is there any example like this where both matrices ##A## and ##B## are hermitian.
Matrices commute when they can be multiplied in any order and still produce the same result. In other words, if A and B are matrices, then AB = BA. This is similar to how real numbers commute under multiplication, but it is not always true for matrices.
No, non-square matrices cannot commute because they cannot be multiplied in both orders. For example, a 2x3 matrix cannot commute with a 3x2 matrix since the first matrix can only be multiplied by a 3x1 or 1x3 matrix, while the second matrix can only be multiplied by a 2x1 or 1x2 matrix.
To determine if two matrices commute, you can multiply them in both orders and see if the resulting matrices are equal. If they are, then the matrices commute. Alternatively, you can check if the matrices share any common eigenvectors, as matrices that share eigenvectors will commute.
An eigenvector is a vector that, when multiplied by a matrix, produces a scalar multiple of itself. In other words, the direction of the vector does not change when multiplied by the matrix. The scalar multiple is called the eigenvalue and represents how much the vector is stretched or compressed by the matrix.
Eigenvectors are used in matrices to simplify calculations involving repeated multiplication of the matrix. By finding the eigenvectors and eigenvalues of a matrix, we can diagonalize it, making it easier to raise to a power or take the logarithm of. Eigenvectors are also used in applications such as image compression and data analysis.