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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.
In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the ...
The trace is a map of Lie algebras : from the Lie algebra of linear operators on an n-dimensional space (n × n matrices with entries in ) to the Lie algebra K of scalars; as K is Abelian (the Lie bracket vanishes), the fact that this is a map of Lie algebras is exactly the statement that the trace of a bracket vanishes: ([,]) =,.
A scalar is an element of a field which is used to define a vector space.In linear algebra, real numbers or generally elements of a field are called scalars and relate to vectors in an associated vector space through the operation of scalar multiplication (defined in the vector space), in which a vector can be multiplied by a scalar in the defined way to produce another vector.
While the first interpretation may be expected by some users due to the nature of implied multiplication, [38] the latter is more in line with the rule that multiplication and division are of equal precedence. [3] When the user is unsure how a calculator will interpret an expression, parentheses can be used to remove the ambiguity. [3]
Hahn–Banach dominated extension theorem [18] (Rudin 1991, Th. 3.2) — If : is a sublinear function, and : is a linear functional on a linear subspace which is dominated by p on M, then there exists a linear extension : of f to the whole space X that is dominated by p, i.e., there exists a linear functional F such that = for all , and | | for ...
In characteristic 2 the latter equality turns into = {, …,} (¯) what therefore provides an opportunity to polynomial-time calculate the Hamiltonian cycle polynomial of any unitary (i.e. such that = where is the identity n×n-matrix), because each minor of such a matrix coincides with its algebraic complement: = (+ /) where ...
Suppose a vector norm ‖ ‖ on and a vector norm ‖ ‖ on are given. Any matrix A induces a linear operator from to with respect to the standard basis, and one defines the corresponding induced norm or operator norm or subordinate norm on the space of all matrices as follows: ‖ ‖, = {‖ ‖: ‖ ‖} where denotes the supremum.