Biomedical Engineering Reference
In-Depth Information
A.3 The Types and Algebra of Square Matrices
The elements of the square matrix
A
given by (A.3) for which the row and column
indices are equal, namely the elements
A
11
,
A
22
,
,
A
nn
, are called diagonal
elements. The sum of the diagonal elements of a matrix is a scalar called the
trace of the matrix and, for a matrix
A
, it is denoted by tr
A
,
...
tr
A
¼
A
11
þ
A
22
þþ
A
nn
:
(A.6)
If the trace of a matrix is zero, the matrix is said to be traceless. Note also that
tr
A
tr
A
T
. A matrix with only diagonal elements is called a diagonal matrix,
¼
2
3
A
11
0
:::
0
4
5
:
0
A
22
:::
0
A
¼
(A.7)
:
:
:
:
:
:
0
:
:
:
:
A
nn
The zero and the unit matrix,
0
and
1
, respectively, constitute the null element,
the 0, and the unit element, the 1, in the algebra of square matrices. The zero matrix
is a matrix whose every element is zero and the unit matrix is a diagonal matrix
whose diagonal elements are all one:
2
3
2
3
00
:::
0
10
:::
0
4
5
;
4
5
:
00
0
::::::
0
:::
01
:::
0
0
¼
1
¼
(A.8)
:
:
:
:
:
:
::::
0
0
:
:
:
:
1
A special symbol,
the Kronecker delta
d
ij
,
is introduced to represent
the
components of the unit matrix. When the indices are equal,
i
¼
j
, the value of the
Kronecker delta is one,
d
11
¼ d
22
¼ ... ¼ d
nn
¼
1 and when they are unequal,
i
0.
The multiplication of a matrix
A
by a scalar is defined as the multiplication of every
element of the matrix
A
by the scalar
6¼
j
, the value of the Kronecker delta is zero,
d
12
¼ d
21
¼ ... ¼ d
n1
¼ d
1n
¼
a
, thus
2
4
3
5
:
a
A
11
a
A
12
:::a
A
1n
a
A
21
a
A
22
:::a
A
2n
a
A
¼
(A.9)
:
:
:
:
:
:
a
A
n1
:
:
:
:
a
A
nn
0
. The
addition of matrices is defined only for matrices with the same number of rows and
columns. The sum of two matrices,
A
and
B
, is denoted by
A
It is then easy to show that 1
A
¼
A
,
1
A
¼
A
,0
A
¼
0
, and
a
0
¼
þ
B
, where
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