§ 18.5
87
BASIC MATHEMATICAL FORMULAS
If the transformation which takes the
into the z; directly is
Xk
n
Zi
= ~ CikX k
is
Zt = CXt,
C = AB and Zt = ABXt
then
(3)
k~1
18.5. Transposed matrix. If A = II a ij II IS m by N, the transposed matrix A' = II a ji II is N by m. For a product C = AB, the transposed matrix is C' = B'A'. For xt, Yt, zt the column matrices of § 18.4 with a single column, the transposed matrices will be row matrices with a single row. We denote them by X, -V, Z. Then the transformations of § 18.4 may be written in terms of row matrices as ~
~
~
Y=XB',
~
~
z= YA',
~
~
~
Z=XC', or Z=XB'A',
sinceC'=B'A'
18.6. Inverse matrix. The unit matrix is a square matrix with ones on the main diagonal and the remaining ele1Dents zero. Its elements are 0ik = 0 if i ~ k, and = 1 if i = k. A square matrix is singular if its determinant is zero. Each nonsingular square matrix has a reciprocal matrix A-I such that AA-l
= Iloik II
and
A-IA
= 110;k II
(1)
Let I A I denote the determinant I aik I, and A ik the cofactor of aik or product of (-1 )i+ k by the determinant obtained from I aik I by striking out the ith row and kth column. (See § 1.8.) Then explicitly A-I has as elements
a
-1
A TAT
_
ki
ik -
(2)
18.7. Symmetry. For square matrices symmetry and skew-symmetry are defined as for tensors in § 7.10. Orthogonal and unitary, or Hermitian orthogonal matrices are defined in § 7.1 1. 18.8. Linear equations. The set of n linear equations in n unknowns
X k,
n
~aik,xk=bi' i=I,2, ... ,n
(1)
k~1
determine
a
Bt = II Bi II
square matrix A = II aile II and two column matrices and xt = II Xi II· In matrix form we m;iy write
AXt
=
Bt
(2)