A Vector space V is a normed space, if there is a norm || || : V -> R + = [ 0, ), defined for elements of V, satisfying:
(1) |||| = 0 <=> =
(2) || + || |||| + |||| , , V
(3) ||a|| = |a| |||| , V, a K ( = R tai C)
Proposition 1. If dim V = n < , then all norms on V are equivalent, i.e., if || || and || ||' are norms defined on V, then a, b > 0:
The set of M of all m x n matrices is a finite dimensional vector space. The following are all norms on M and A = (aij ) is an m x n matrix.
||A||1 = | aij | , the maximum of column sums,
||A|| = | aij | , the maximum of row sums,
||A||2 = (the spectral norm), is the largest eigenvalue of the matrix TA, and
||A||Fr = , Frobenius norm,
which all are special cases of a norm or the form
(4) ||A|| = [ ||A|| / |||| ],
where || || is a norm defined on Kn.
Example 1: Norms of a matrix
Note. All the norms defined above satisfy ||AB|| ||A|| ||B||.
In numerical computations one often needs the condition number of a regular matrix A:
A sequence 1 , ..., n , ... of vectors of V converges to V in the norm || || if
The convergence a sequence of matrices is defined by the same manner.
It follows from Proposition 1 that if dim V = n < (in particular, if V = Rn, M) and if n -> with respect to a norm || ||, then n -> also in every other norm defined on V (equivalence).
A matrix series Ak , Ak M, converges (in a matrix norm) if the sequence of partial sums
converges. It can be shown that Ak = (a (k)ij ) -> A = (aij ) <=> a (k)ij -> aij i,j kun k -> .
Let A be a square matrix. We define the matrix eA by setting
If A is diagonalizable, i.e., if A = TDT - 1 where D = diag (1 , ..., n ), then
|Ak||= TDkT - 1||= Tdiag (1 k, ..., kn )T - 1||=>
||(1 / k!) Ak
( (1 k / k!), ..., (n k / k!) )T - 1|| =
|| T [ diag (
||(1 k / k! ), ..., (n k / k! ) ) ] T - 1|
it follows that
||| ( 1 / k!) Ak - TeDT -1|||| = ||||T [ diag ( (1k / k!) , ..., (nk / k!) ) ] T -1
- TeDT -1|||| = ||||T [ diag ( (1k / k!) , ..., (nk / k!) ) - eD ] T -1||||||T|| || diag ( (1k / k!) , ..., (nk / k!) ) - eD|| ||T -1|||| ->|| 0, kun k -> .|
Hence: If A = TDT - 1, then
The solution of the differential equation '(t) = A(t) (A does not need to diagonalizable) with an initial condition (0) = 0 is
etA is the so-called state-transition matrix. If kontrolli u(t) on mukana eli kun
(B is a column vector).
Example 2: eA