Robert1986 said:Do you know that symmetric matrices can be (orthogonally) diaganalised?
Repeated eigenvalues of a symmetric matrix are eigenvalues that have a multiplicity greater than one. This means that there are multiple linearly independent eigenvectors associated with the same eigenvalue.
Repeated eigenvalues can make the diagonalization of a symmetric matrix more complicated as they may result in fewer distinct eigenvalues and therefore fewer linearly independent eigenvectors. This can lead to a diagonal matrix with repeated entries along the diagonal.
Yes, a symmetric matrix can have repeated eigenvalues. In fact, all symmetric matrices have at least one repeated eigenvalue, which is equal to the matrix's trace.
The geometric interpretation of repeated eigenvalues is that they represent directions in which the matrix has the same effect on all vectors. This means that the matrix has a higher degree of symmetry in these directions.
Repeated eigenvalues do not affect the spectral theorem for symmetric matrices. The spectral theorem states that a symmetric matrix can be diagonalized by an orthogonal matrix, and this remains true even if the matrix has repeated eigenvalues.