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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.
A canonical form may simply be a convention, or a deep theorem. For example, polynomials are conventionally written with the terms in descending powers: it is more usual to write x 2 + x + 30 than x + 30 + x 2, although the two forms define the same polynomial. By contrast, the existence of Jordan canonical form for a matrix is a deep theorem.
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.
In Boolean algebra, any Boolean function can be expressed in the canonical disjunctive normal form , [1] minterm canonical form, or Sum of Products (SoP or SOP) as a disjunction (OR) of minterms. The De Morgan dual is the canonical conjunctive normal form ( CCNF ), maxterm canonical form , or Product of Sums ( PoS or POS ) which is a ...
Then Q is a torsion-free Z-module which is not free. Another classical example of such a module is the Baer–Specker group, the group of all sequences of integers under termwise addition. In general, the question of which infinitely generated torsion-free abelian groups are free depends on which large cardinals exist.
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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 Weyr canonical form = is related to the Jordan form by a simple permutation for each Weyr basic block as follows: The first index of each Weyr subblock forms the largest Jordan chain. After crossing out these rows and columns, the first index of each new subblock forms the second largest Jordan chain, and so forth.