1.1 Basic Concepts and Formulae 13
Matrices
Types of matrices and definitions
Identity matrix:
I 2 =
(
10
01
)
;I 3 =
⎛
⎝
100
010
001
⎞
⎠ (1.78)
Scalar matrix:
(
a 11 0
0 a 22)
;
⎛
⎝
a 11 00
0 a 22 0
00 a 33⎞
⎠ (1.79)
Symmetric matrix:
(
aji=aij)
;
⎛
⎝
a 11 a 12 a 13
a 12 a 22 a 23
a 13 a 23 a 33⎞
⎠ (1.80)
Skew symmetric:
(
aji=−aij)
;
⎛
⎝
a 11 a 12 a 13
−a 12 a 22 a 23
−a 13 −a 23 a 33⎞
⎠ (1.81)
TheInverse of a matrix B=A−^1 (B equals A inverse):
ifAB=BA=Iand further, (AB)−^1 =B−^1 A−^1
Acommutes withBifAB=BA
Aanti-commutes withBifAB=−BA
TheTranspose(A′)of a matrix Ameans interchanging rows and columns.
Further,(A+B)′=A′+B′
(A′)′=A,(kA)′=kA′ (1.82)TheConjugate of a matrix. If a matrix has complex numbers as elements, and if
each number is replaced by its conjugate, then the new matrix is called the conjugate
and denoted byA∗orA(Aconjugate)
TheTrace (Tr) or Spurof a matix is the num of the diagonal elements.
Tr=∑
aii (1.83)