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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 .
The term diagonal matrix may sometimes refer to a rectangular diagonal matrix, which is an m-by-n matrix with all the entries not of the form d i,i being zero. For example: [ 1 0 0 0 4 0 0 0 − 3 0 0 0 ] or [ 1 0 0 0 0 0 4 0 0 0 0 0 − 3 0 0 ] {\displaystyle {\begin{bmatrix}1&0&0\\0&4&0\\0&0&-3\\0&0&0\\\end{bmatrix}}\quad {\text{or}}\quad ...
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.
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.
This makes normal operators, and normal elements of C*-algebras, more amenable to analysis. The spectral theorem states that a matrix is normal if and only if it is unitarily similar to a diagonal matrix, and therefore any matrix A satisfying the equation A * A = AA * is diagonalizable. (The converse does not hold because diagonalizable ...
The definition in the first paragraph sums entries across each row. It is therefore sometimes called row diagonal dominance. If one changes the definition to sum down each column, this is called column diagonal dominance. Any strictly diagonally dominant matrix is trivially a weakly chained diagonally dominant matrix.
Let () (that is, a n × n complex matrix) and () be the change of basis matrix to the Jordan normal form of A; that is, A = C −1 JC.Now let f (z) be a holomorphic function on an open set such that ; that is, the spectrum of the matrix is contained inside the domain of holomorphy of f.