Is the Markov matrix always symmetrical

Eigenvalues ​​and Eigenvectors

Mathematics for Computer Scientists pp 385-410 | Cite as

Part of the eXamen.press book series (EXAMS)

abstract

We have a lot of n linearly independent vectors u1, . . . , un ∈ ℝn (or ℂn) is called the base, since every vector x un ∈ ℝn as a linear combination
$$ {\ text {x =}} \ sum \ limits_ {j {\ text {= 1}}} ^ n {y_j u_j} $$
lets write. If we consider these base vectors to be fixed, the vector x can be given both by its Coordinatesx1, . . . , xn regarding the standard base e1,. . . , en, as well as by its coordinates y1, . . . , yn regarding the new base u1, . . . , un to be discribed. If we use the basis vectors uj as columns of a matrix
$$ U = (u_1 u_2 ... u_n) $$
then we can easily calculate back and forth between the different coordinates:
$$ x = Uy, y = U ^ {- 1} x. $$
In particular, the relationship between the basis vectors is through
$$ u_j = Ue_j, j = 1, ..., n $$
given, i.e., the matrix U is the matrix of that linear map that contains the old base e1,. . . , en in the new base u1, . . . , un convicted. This linear mapping or the associated matrix U becomes Base transformation or Coordinate transformation called.
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© Springer-Verlag Berlin Heidelberg 2008