- #1
Uncle_John
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Is it possible to diagonalize such matrix? and how would one do it?
Diagonalization of a square matrix is the process of finding a diagonal matrix that is similar to the original matrix. This means that the diagonal matrix has the same eigenvalues and eigenvectors as the original matrix, but with the additional property that all non-diagonal elements are equal to zero.
Diagonalization is important because it simplifies the calculation of powers and determinants of a matrix. It also allows us to easily solve systems of linear equations and find the inverse of a matrix.
If not all eigenvalues are distinct, it means that some eigenvalues have multiplicity greater than one. This means that there are multiple eigenvectors associated with the same eigenvalue.
Yes, a matrix can still be diagonalized even if not all eigenvalues are distinct. However, the diagonal matrix obtained will not be unique and may not have all zero non-diagonal elements.
If not all eigenvalues are distinct, we can still use the same process of finding eigenvectors and forming a similarity matrix to diagonalize the matrix. However, in this case, the diagonal matrix may not have all zero non-diagonal elements and may not be unique.