Computational Physics - Department of Physics

154 6 Linear Algebra

Various matrix and vector operations are provided through optional integration with BLAS

and LAPACK.

6.2 Mathematical Intermezzo

The matrices we will deal with are primarily square real symmetric or hermitian ones, assum-

ing thereby that ann×nmatrixA∈Rn×nfor a real matrix^1 andA∈Cn×nfor a complex matrix.

For the sake of simplicity, we take a matrixA∈R^4 ×^4 and a corresponding identity matrixI

A=







a 11 a 12 a 13 a 14

a 21 a 22 a 23 a 24

a 31 a 32 a 33 a 34

a 41 a 42 a 43 a 44





 I=







1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1





, (6.1)

whereai j∈R. The inverse of a matrix, if it exists, is defined by

A−^1 ·A=I.

In the following discussion, matrices are always two-dimensional arrays while vectors are

one-dimensional arrays. In our nomenclature we will restrict boldfaced capitals letters such

asAto represent a general matrix, which is a two-dimensional array, whileai jrefers to a

matrix element with row numberiand column number j. Similarly, a vector being a one-

dimensional array, is labelledxand represented as (for a real vector)

x∈Rn⇐⇒







x 1

x 2

x 3

x 4





,

with pertinent vector elementsxi∈R. Note that this notation impliesxi∈R^4 ×^1 and that the

members ofxare column vectors. The elements ofxi∈R^1 ×^4 are row vectors.

Table 6.2 lists some essential features of various types of matrices one may encounter.

Some of the matrices we will encounter are listed here

Table 6.1Matrix properties

Relations Name matrix elements

A=AT symmetric ai j=aji

A=

(

AT

)− 1

real orthogonal∑kaikajk=∑kakiak j=δi j

A=A∗ real matrix ai j=a∗i j

A=A† hermitian ai j=a∗ji

A=

(

A†

)− 1

unitary ∑kaika∗jk=∑ka∗kiak j=δi j

1. Diagonal ifai j= 0 fori 6 =j,

(^1) A reminder on mathematical symbols may be appropriate here.The symbolRis the set of real numbers.
Correspondingly,N,ZandCrepresent the set of natural, integer and complex numbers, respectively. A symbol
likeRnstands for ann-dimensional real Euclidean space, whileC[a,b]is the space of real or complex-valued
continuous functions on the interval[a,b], where the latter is a closed interval. Similalry,Cm[a,b]is the space
ofm-times continuously differentiable functions on the interval[a,b]. For more symbols and notations, see the
main text.