Search results
Results From The WOW.Com Content Network
This real Jordan form is a consequence of the complex Jordan form. For a real matrix the nonreal eigenvectors and generalized eigenvectors can always be chosen to form complex conjugate pairs. Taking the real and imaginary part (linear combination of the vector and its conjugate), the matrix has this form with respect to the new basis.
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.
Rather, the Jordan canonical form of () contains one Jordan block for each distinct root; if the multiplicity of the root is m, then the block is an m × m matrix with on the diagonal and 1 in the entries just above the diagonal. in this case, V becomes a confluent Vandermonde matrix. [2]
Jordan canonical form — an 'almost' diagonalised matrix, where the only non-zero elements appear on the lead and superdiagonals. Linear independence — two or more vectors are linearly independent if there is no way to construct one from linear combinations of the others. Matrix exponential — defined by the exponential series.
Jordan normal form is a canonical form for matrix similarity. The row echelon form is a canonical form, when one considers as equivalent a matrix and its left product by an invertible matrix . In computer science, and more specifically in computer algebra , when representing mathematical objects in a computer, there are usually many different ...
If they are chosen in a particularly judicious manner, we can use these vectors to show that is similar to a matrix in Jordan normal form. In particular, Definition: A set of n linearly independent generalized eigenvectors is a canonical basis if it is composed entirely of Jordan chains.
This form is a special case of the Jordan canonical form for matrices. [7] For example, any nonzero 2 × 2 nilpotent matrix is similar to the matrix []. That is, if is any nonzero 2 × 2 nilpotent matrix, then there exists a basis b 1, b 2 such that Nb 1 = 0 and Nb 2 = b 1
Jordan normal form — bidiagonal matrix of a certain form; generalizes the eigendecomposition Weyr canonical form — permutation of Jordan normal form; Jordan–Chevalley decomposition — sum of commuting nilpotent matrix and diagonalizable matrix; Schur decomposition — similarity transform bringing the matrix to a triangular matrix