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The diagonal form for diagonalizable matrices, for instance normal matrices, is a special case of the Jordan normal form. [6] [7] [8] The Jordan normal form is named after Camille Jordan, who first stated the Jordan decomposition theorem in 1870. [9]
Indeed, determining the Jordan normal form is generally a computationally challenging task. From the vector space point of view, the Jordan normal form is equivalent to finding an orthogonal decomposition (that is, via direct sums of eigenspaces represented by Jordan blocks) of the domain which the associated generalized eigenvectors make a ...
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]
The Jordan normal form and the Jordan–Chevalley decomposition. Applicable to: square matrix A; Comment: the Jordan normal form generalizes the eigendecomposition to cases where there are repeated eigenvalues and cannot be diagonalized, the Jordan–Chevalley decomposition does this without choosing a basis.
A matrix normal form or matrix canonical form describes the transformation of a matrix to another with special properties. Pages in category "Matrix normal forms" The following 10 pages are in this category, out of 10 total.
Find a matrix in Jordan normal form that is similar to = (). Solution: The characteristic equation of is () =, hence, = is an eigenvalue of algebraic multiplicity three. Following the procedures of the previous sections, we find that
An example of a matrix in Jordan normal form. The grey blocks are called Jordan blocks. The eigendecomposition or diagonalization expresses A as a product VDV −1, where D is a diagonal matrix and V is a suitable invertible matrix. [52] If A can be written in this form, it is called diagonalizable.
The Drazin inverse is then the operation that maps invertible Jordan blocks to their inverses, and nilpotent Jordan blocks to zero. More generally, we may define the Drazin inverse over any perfect field , by using the Jordan-Chevalley decomposition A = A s + A n {\displaystyle A=A_{s}+A_{n}} where A s {\displaystyle A_{s}} is semisimple and A ...