Jordan standard format
the Jordan standard format is a term from the mathematical subsection of linear algebra. She is a simple representative of the equivalence class of the stencils similar to a triangulierbaren matrix (triangulierbaren linear illustration ) (linear illustrations). The Triangulierbarkeit is equivalent with the fact that the characteristic polynomialcompletely into linear factors disintegrates to the matrix (linear illustration ). The Jordan standard format was designated after Marie Ennemond Camille Jordan.
The Jordan standard format to a square matrix <math> A< /math> a matrix is <math> J< /math> in the following block diagonal form
- < math> J= \ begin {pmatrix}
J_1 & & 0 \ \
& \ ddots& \ \ 0 & & J_k \ end {pmatrix}
= Q^ {- to 1} AQ< /math>
The matrix Q is the matrix of the self-vectors and main vectors, i.e. the self-vectors and pertinent main vectors are written to Q -1 in columns as matrix, are the inverse matrix of Q.
The J i are the Jordan blocks.These have the following form:
<math> J_i= \ begin {pmatrix} \ lambda_i & 1 & & & 0 \ \ & \ lambda_i & 1 & & \ \ && \ ddots {} & \ ddots {} \ \ &&& \ lambda_i & 1 \ \ 0 & & & & \ lambda_i \ end {pmatrix}< to /math>
Λ the i are thereby the eigenvaluesof A. To each eigenvalue λ i gives accordingly it to its geometrical multiplicity many Jordan blocks. ( The geometrical multiplicity is certain thereby by the number of linear independent self-vectors to one (!) Eigenvalue λ.) the total dimension of all Jordan blocks of an eigenvalue corresponds to its algebraic multiplicity, i.e. its multiplicityin the characteristic polynomial.
In a Jordan block the so-called Jordan chains are “stored” (see main vector). Math <A> /math< exists> e.g. only from a Jordan block with eigenvalue <math> \ lambda< /math> and designate <math> e_j< /math> <math> the j< /math> - it ten unit vector then one sees easy that <math> (A \ lambda E) e_1 = 0< /math> and <math> (A \ lambdaE) e_ {j+1} = e_j< /math> for <math> j=1, \ dots, n-1< /math> applies.
In the special case of a diagonalisierbaren matrix the Jordan' standard format is a diagonal matrix.
to the form from Q
are <math> \ vec e_i </math> the self-vectors to the Jordan small boxes <math> J_i< /math> (Corresponds to each Jordan small box exactly a self-vector) and <math> \ vec h_ {i, 2} \ dots {}\ vec h_ {i, l} \ dots {} \ vec h_ {i, n_j}< /math> the main vectors that l-ten in each case stage, whereby n j the dimension j-ten Jordan small box is.
The matrix Q has then the form:
<math> Q = \ vec e_1, \ vec h_ {1.2}, \ dots {}, \ vec h_ {1, n_1}, \ dots {}, \ vec e_k,\ vec h_ {k, 2}, \ dots {}, \ vec h_ {k, n_k} </math>
whereby k was the number of Jordan small boxes.
In words: The columns of Q are the self-vectors with the pertinent main vectors in the order of the pertinent Jordan small boxes.
real Jordan standard format
regards one stencils over that, then it can happen to real numbers R the fact that the characteristic polynomial does not disintegrate thus into linear factors (possesses complex zeros). Since one can not use the normal Jordan standard format in this case (it would have entries from C), introduces one the real Jordan form.
Is <math> \ lambda_j = a_j + b_j i </math> Eigenvalue then is also
< math> \ bar \ lambda_j = a_j - b_j i </math> Eigenvalue
whereby <math> i^2 = -1< /math>
The J j of Jordan blocks have the now following form:
<math> J_j= \ begin {pmatrix} a_j & b_j & 1 & & & && 0 \ \ - b_j & a_j & 0 & 1& & & & \ \ & & a_j & b_j & 1 & & & \ \ & & - b_j & a_j & 0 & 1& & \ \ &&& \ ddots {} & \ ddots {} & \ ddots {} & \ ddots {} &\ \ &&&& \ ddots {} & \ ddots {} & 0 & 1 \ \ &&&&& \ ddots {} & a_j & b_j \ \ 0 &&&&& & - b_j & a_j \ \
\ end {pmatrix}< to /math>
to the matrix J applies still:
<math>
J= \ begin {pmatrix} J_1 & & 0 \ \ & \ ddots {} & \ \ 0 && J_k \ end {pmatrix} = Q^ {- to 1} AQ
< /math>
However for another Q
of systems of linear differential equations
the JNF is closely linked with linear differential equations y'(x) =Ay (x) in a dimension n, i.e. A is one n×n - matrix with constant real or complex components. This hasthe formal solution by power series beginning
- <math> y (x) = \ sum_ {k=0} ^ \ infty \ frac {A^kx^k} {k!}y_0=y_0+x \ cdot Ay_0+ \ frac {x^2} {2} \ cdot A^2y_0+ \ dots< /math>
with initial value y (0) =y 0.
Now if the equation applies math for A^my_0=0 /math to <any> exponent< m>, then the power series breaks, the system off has a polynomiale solution of the degree of small m. One knows now the cores of the matrix powersintends, for these applies <math> \ ker A \ subset \ ker A^2 \ subset \ dots \ subset \ ker A^n< /math>. If one determines now to this box sequence of Unterräumen adapted a basis of <math> \ ker A^n< /math>, then one receives the Jordan blocks to the eigenvalue λ=0. For example A m+1 v=0, but A m v is von Nulldifferently, then math <v>, Av, \ dots, are A^m v< /math> linear independently and the box sequence adapted.
For the general case one makes the beginning <math> for y (x) = \ exp (\ lambda x) u (x)< /math>, it arises for u the differential equation <math> u' = (A \ lambda I) u< /math>. Thus the maximum core <math> \ more ker (A \ lambda I) ^n< /math> is nontrivial,λ eigenvalue must, i.e. Zero of the characteristic polynomial December (A-λI), of A its. The cores of different eigenvalues are transverse to each other, so that a common, which respective cores adapted basis of the entire n - dimansionalen area to be found can. For each basis element a solution results<math> y (x) =exp (\ lambda x) u (x)< /math> with polynomialem u, and each solution can be built up from these (superposition principle). After suitable sorting of the adapted basis and transformation of A into this basis the Jordan standard format results.
