# Matrices commute & Eigenvectors question

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• LagrangeEuler
In summary, it is possible to find matrices that commute but have different eigenvectors. For example, matrix A = [0,1;0,0] and B = [1,0;0,1] commute, but B has an eigenvector (0,1) that A does not have. It is also possible to find two hermitian matrices A = I_n and B, where B is hermitian, that commute but have different eigenvectors.
LagrangeEuler
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?

##A = \pmatrix{0&1\\0&0}, B = \pmatrix{1&0\\0&1}## commute. ##B## has an eigenvector ##(0,1)##, which ##A## doesn't have. They do have a common eigenvector.

EDIT: Take ##A = I_n## and ##B \in M_{n,n}(\mathbb{F})## where ##\mathbb{F}## is a field. Then you can see that this claim is obviously false.

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?

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?

Matrix A does have eigenvectors.. (but yes, A is not diagonalizable because we do not have enough linearly independent eigenvectors)

##det(A - \lambda I_n) = 0 \iff \lambda = 0##

Thus ##0## is an eigenvalue. We deduce that the eigenvectors are ##V_0 = span(\{(1,0)\})##

As for your question with the hermitian matrices, look at the edit in my reply to your first post.

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Thanks a lot. Is it possible to construct similar example for to Hermitian matrices?

Yes, look at the edit in my first post. (The identity matrix is hermitian)

Math_QED said:
Yes, look at the edit in my first post. (The identity matrix is hermitian)
Yes. But the other one is not. Is there any example like this where both matrices ##A## and ##B## are hermitian.

LagrangeEuler said:
Yes. But the other one is not. Is there any example like this where both matrices ##A## and ##B## are hermitian.

Take ##A = I_n## and ##B## such that ##B## is hermitian. You should do some effort yourself (almost any hermitian matrix for ##B## will give you what you search for).

fresh_42

## 1. What does it mean for matrices to commute?

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.

## 2. Can non-square matrices commute?

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.

## 3. How can I determine if two matrices commute?

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.

## 4. What is an eigenvector?

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.

## 5. How are eigenvectors used in matrices?

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.

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