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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 .
This is because any function of a non-defective matrix acts directly on each of its eigenvalues, and the conjugate transpose of its spectral decomposition is , where is the diagonal matrix of eigenvalues. Likewise, if two normal matrices commute and are therefore simultaneously diagonalizable, any operation between these matrices also acts on ...
Let A be a square n × n matrix with n linearly independent eigenvectors q i (where i = 1, ..., n).Then A can be factored as = where Q is the square n × n matrix whose i th column is the eigenvector q i of A, and Λ is the diagonal matrix whose diagonal elements are the corresponding eigenvalues, Λ ii = λ i.
An n × n matrix A is diagonalizable if and only if the sum of the dimensions of the eigenspaces is n. Or, equivalently, if and only if A has n linearly independent eigenvectors. Not all matrices are diagonalizable; matrices that are not diagonalizable are called defective matrices. Consider the following matrix:
[4]: p. 64 The converse is also true; that is, if two diagonalizable matrices commute, they are simultaneously diagonalizable. [5] But if you take any two matrices that commute (and do not assume they are two diagonalizable matrices) they are simultaneously diagonalizable already if one of the matrices has no multiple eigenvalues. [6]
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 {,}.
If is diagonalizable it may be decomposed as = where is an orthogonal matrix =, and is a diagonal matrix of the eigenvalues of . In the special case that A {\displaystyle A} is real symmetric, then Q {\displaystyle Q} and Λ {\displaystyle \Lambda } are also real.
2. The upper triangle of the matrix S is destroyed while the lower triangle and the diagonal are unchanged. Thus it is possible to restore S if necessary according to for k := 1 to n−1 do ! restore matrix S for l := k+1 to n do S kl := S lk endfor endfor. 3. The eigenvalues are not necessarily in descending order.