Page 91 - Matrix Analysis & Applied Linear Algebra

3.2 Addition and Transposition 85

T T T
This proves that corresponding entries in (A + B) and A + B are equal,
T T T
so it must be the case that (A + B) = A + B . Similarly, for each i and j,
T T T T
[(αA) ] ij =[αA] ji = α[A] ji = α[A ] ij =⇒ (αA) = αA .
Sometimes transposition doesn’t change anything. For example, if
 
123
T
A =  245  , then A = A.
356
This is because the entries in A are symmetrically located about the main di-
agonal—the line from the upper-left-hand corner to the lower-right-hand corner.
 0 0 
λ 1 ···
0 λ 2 ··· 0
Matrices of the form D =  . . . .  are called diagonal matrices,
. . . .
. . . .
0 0 ··· λ n
T
and they are clearly symmetric in the sense that D = D . This is one of several
kinds of symmetries described below.
Symmetries
Let A =[a ij ] be a square matrix.

T
• A is said to be a symmetric matrix whenever A = A , i.e.,
whenever a ij = a ji .

T
• A is said to be a skew-symmetric matrix whenever A = −A ,
i.e., whenever a ij = −a ji .
∗
• A is said to be a hermitian matrix whenever A = A , i.e.,
whenever a ij = a ji . This is the complex analog of symmetry.

∗
• A is said to be a skew-hermitian matrix when A = −A , i.e.,
whenever a ij = −a ji . This is the complex analog of skew symmetry.

For example, consider
   
1 2+4i 1 − 3i 1 2+4i 1 − 3i
A =  2 − 4i 3 8+6i  and B =  2+4i 3 8+6i  .
1+3i 8 − 6i 5 1 − 3i 8+6i 5
Can you see that A is hermitian but not symmetric, while B is symmetric but
not hermitian?
Nature abounds with symmetry, and very often physical symmetry manifests
itself as a symmetric matrix in a mathematical model. The following example is
an illustration of this principle.