Two matrices having the same eigenspaces means that they have the same set of eigenvectors and eigenvalues. This means that when a vector is multiplied by either matrix, the resulting vector will be in the same direction (eigenvector) and have the same scaling factor (eigenvalue).
One example is the identity matrix and its transpose. The identity matrix has all 1's on the main diagonal and 0's everywhere else, and its transpose is just itself. Since the identity matrix has 1 as its only eigenvalue and all vectors are eigenvectors, its transpose will have the same eigenspaces.
To prove that two matrices have the same eigenspaces, you can show that they have the same eigenvalues and corresponding eigenvectors. This can be done by finding the eigenvalues of each matrix and then solving for the corresponding eigenvectors. If the eigenvalues and eigenvectors match, then the matrices have the same eigenspaces.
No, if two matrices have the same eigenspaces, they must have the same eigenvalues. This is because the eigenvalues determine the scaling factor for the eigenvectors, so if the eigenspaces are the same, the eigenvalues must also be the same.
If A and A transpose have the same eigenspaces, it means that these matrices have some special properties. For example, they will have the same determinant, trace, and rank. This can also be useful in solving certain problems, such as finding the diagonalization or eigenvalues of a matrix.