Linear Algebra
In the following only real vectors and matrices are considered.
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4 Linear Algebra 4.1 Matrices Basic concepts In the following only real vectors and matrices are considered.
Colunm vectora= [
::l
E
Row vecW a T= (ai' "', a.)
R·
Scalar product aTb=aIb I + ... +anbn Norm (length)
lal
Jr~-i-+-..-.-+-a-~
= Ja Ta =
Matrix of order mXn: (A is square if m=n):
1
a Ij aIn] = (aij) = [aI' ... , aj' ... , an]' aj= [ ... A= [all ... amI··· a mn a mj Transpose of A: A T=
I~~.I
... amI] of order nXm (exchange rows and columns).
laIn ... a mn all 0 ... 0 . . 0 a22 ... 0 [ Dzagonal matnx D= ... . . . OO
j
. =dlag(all, ... , ann)
(aij=O, i"#j)
... Oann
j
Identity matrix I=diag(1, 1, ... , 1) of order nXn
all 0 0 a2I a22 0 Lower triangular matrix T = ...
r
an Ia n2
... 0 ... 0
(aij=O, iPj if iO.
Decomposition of matrices 29. For any square matrix A there exist unique Hermitian matrices HI and H2 [H] =(A+A*)12 and H] = (A-A*)l2i] such thatA=H] +iH2 .
30. N is normal N=H I +iH2 with commuting Hermitian matrices H] and H2 (i.e. H]H2=H2H]). 31. Let H] and H2 be Hermitian. Then there exists a unitary matrix U simultaneously diagonalizing H] and H2 (i.e. U*H] U and U*H2U are diagonal) H]H2 =H2H].
Non-unitary transformations 32. Assume that the square nxn-matrix A has n linear independent eigenvectors gl, g2, ... , gn (e.g. this is the case if the n eigenvalues A], A2' ... , An are distinct). Then with L= [g], g2, ... , gn], C]AL=D=diag(A], A2, ... , An)·
33. Multiple eigenvalues. Generalized eigenvectors. The vector v;t 0 is a generalized eigenvector corresponding to an eigenvalue A of multiplicity k of the nxn-matrix A, if (A - A/)kv = O. To an eigenValue of multiplicity k there always correspond k linearly independent generalized eigenvectors. Generalized eigenvectors corresponding to different eigenvalues are linearly independent.
34. Jordan form: For any square matrix A there exists a non-singular matrix S such that
o ........
Ai 1 0 0 Ai 1 0 ....... 0
0
J2 0 ...... 0
o , Ji =
................ 0 Jm
o ........ 0 Ai o .............. 0
Ai
where Ji are the Jordan blocks. The same eigenvalue Ai may appear in several blocks if it corresponds to several independent eigenvectors. The columns of S are generalized eigenvectors of A and constitute a basis of
en.
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