Biology Reference
In-Depth Information
If A is a square matrix (i.e., has the same number of rows as columns),
then sometimes there is such a matrix A 1 , called the inverse of A. Thus,
we somehow need to create a square matrix in Eq. (8-62) in order to
solve for
e
. We now describe how to do this.
Associated with each matrix A is its transposed matrix A T . The matrix A T
is obtained by forming the matrix whose rows are the columns of A.
0
1
. Notice, if A
14
25
36
123
456
@
A ;
then A T
Thus if, for example, A
¼
¼
n matrix, then A T is an n
m matrix, so A T A (n
is an m
m multiplied
n matrix, and AA T (m
by an m
n)isann
n multiplied by an n
m
matrix) is an m
m matrix. Thus, either product is a square matrix.
There is a possibility that (A T A) 1 exists (again, computers will routinely
check this), and, if so, we can solve in the following way. First multiply
both sides of the equation by A T from the left to get
A T A
A T b
e ¼
:
(8-63)
Next, find the inverse (A T A) 1 and multiply both sides of Eq. (8-63) by
this matrix from the left to obtain the vector of the unknowns
e
:
Þ 1
Þ 1 A T b
A T A
A T A
A T A
ð
ð
Þe ¼ð
Þ 1 A T b
A T A
e ¼ð
:
(8-64)
Therefore, if the inverse matrix (A T A) 1 exists, the solution
e
of Eq. (8-62)
is given by Eq. (8-64).
REFERENCES
Efron, B., & Tibshirani, R. J. (1993). An introduction to the bootstrap. New York:
Chapman and Hall.
Johnson, M. L., & Frasier, S. G. (1985). Nonlinear least-squares analysis. In Hirs,
C. H. W., & Timasheff, S. N. (eds.), Methods in Enzymology (vol. 117, pp.
301-342). New York: Academic Press.
Straume, M., & Johnson, M. L. (1992). Analysis of residuals: Criteria for
determining goodness-of-fit. In Brand, L., & Johnson, M. L. (eds.), Methods in
Enzymology (vol. 210, pp. 87-105). New York: Academic Press.
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