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A first-order matrix difference equation with constant term can be written as + = +, where A is n × n and y and c are n × 1.This system converges to its steady-state level of y if and only if the absolute values of all n eigenvalues of A are less than 1.
An essential question in linear algebra is testing whether a linear map is an isomorphism or not, and, if it is not an isomorphism, finding its range (or image) and the set of elements that are mapped to the zero vector, called the kernel of the map.
In linear algebra, two n-by-n matrices A and B are called similar if there exists an invertible n-by-n matrix P such that =. Similar matrices represent the same linear map under two (possibly) different bases , with P being the change-of-basis matrix .
The rank–nullity theorem is a theorem in linear algebra, which asserts: the number of columns of a matrix M is the sum of the rank of M and the nullity of M ; and the dimension of the domain of a linear transformation f is the sum of the rank of f (the dimension of the image of f ) and the nullity of f (the dimension of the kernel of f ).
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: ([,]) =,.
Numerical linear algebra, sometimes called applied linear algebra, is the study of how matrix operations can be used to create computer algorithms which efficiently and accurately provide approximate answers to questions in continuous mathematics. It is a subfield of numerical analysis, and a type of linear algebra.