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The fundamental fact about diagonalizable maps and matrices is expressed by the following: An matrix over a field is diagonalizable if and only if the sum of the dimensions of its eigenspaces is equal to , which is the case if and only if there exists a basis of consisting of eigenvectors of .
In fact, a given n-by-n matrix A is similar to a diagonal matrix (meaning that there is a matrix X such that X −1 AX is diagonal) if and only if it has n linearly independent eigenvectors. Such matrices are said to be diagonalizable .
An idempotent matrix is always diagonalizable. [3] Its eigenvalues are either 0 or 1: if is a non-zero eigenvector of some idempotent matrix and its associated eigenvalue, then = = = = =, which implies {,}.
A square matrix that does not have a complete basis of eigenvectors, and is thus not diagonalizable. Derogatory matrix: A square matrix whose minimal polynomial is of order less than n. Equivalently, at least one of its eigenvalues has at least two Jordan blocks. [3] Diagonalizable matrix: A square matrix similar to a diagonal matrix.
The roots of the characteristic polynomial () are the eigenvalues of ().If there are n distinct eigenvalues , …,, then () is diagonalizable as () =, where D is the diagonal matrix and V is the Vandermonde matrix corresponding to the λ 's: = [], = [].
An n × n matrix commutes with every other n × n matrix if and only if it is a scalar matrix, that is, a matrix of the form , where is the n × n identity matrix and is a scalar. In other words, the center of the group of n × n matrices under multiplication is the subgroup of scalar matrices.
A semi-simple matrix is one that is similar to a direct sum of simple matrices; if the field is algebraically closed, this is the same as being diagonalizable. These notions of semi-simplicity can be unified using the language of semi-simple modules, and generalized to semi-simple categories.
In logic and mathematics, diagonalization may refer to: Matrix diagonalization, a construction of a diagonal matrix (with nonzero entries only on the main diagonal) that is similar to a given matrix; Diagonal argument (disambiguation), various closely related proof techniques, including: