Page 93 - Matrix Analysis & Applied Linear Algebra

3.2 Addition and Transposition 87

Organize the above three equations as a linear system:

k 1 x 1 − k 1 x 2 = F 1 ,
−k 1 x 1 +(k 1 + k 2 )x 2 − k 2 x 3 = F 2 ,
−k 2 x 2 + k 2 x 3 = F 3 ,
and observe that the coefficient matrix, called the stiffness matrix,
 
k 1 −k 1 0
K =  −k 1 k 1 + k 2 −k 2  ,
0 −k 2 k 2
is a symmetric matrix. The point of this example is that symmetry in the physical
problem translates to symmetry in the mathematics by way of the symmetric
matrix K. When the two springs are identical (i.e., when k 1 = k 2 = k ), even
more symmetry is present, and in this case
 
1 −1 0
K = k  −1 2 −1  .
0 −1 1

Exercises for section 3.2

3.2.1. Determine the unknown quantities in the following expressions.

T
03 x +2 y +3 3 6
(a) 3X = . (b) 2 = .
69 3 0 y z
3.2.2. Identify each of the following as symmetric, skew symmetric, or neither.
   
1 −3 3 0 −3 −3
(a)  −3 4 −3  . (b)  3 0 1  .
3 3 0 3 −1 0
 
0 −3 −3
120
(c)  −3 0 3  . (d) .
210
−3 3 1
3.2.3. Construct an example of a 3 × 3 matrix A that satisfies the following
conditions.
(a) A is both symmetric and skew symmetric.
(b) A is both hermitian and symmetric.
(c) A is skew hermitian.