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3/22/2011
Math Modeling, I (GAUSS project)
1
LINEAR ALGEBRA REVIEW
Math Modeling I, Spring 2011
Linear Algebra
•
View a matrix as an “array of columns”:
A
mxn
= [
a
1
,
a
2
,…,
a
n
] with
a
i
=[a
1i
,a
2i
,…,a
mi
]
T
•
If x=[x
1
,x
2
,…,x
n
]
T
, then Ax can be viewed as a linear
combination of the columns of A:
Ax = x
1
a
1
+x
2
a
2
+…+x
n
a
n
•
A matrix is symmetric if A=A
T
•
A matrix is orthogonal if its columns are orthogonal
vectors:
a
i
T
a
j
=0 for all i
≠
j;
a
i
T
a
i
=1.
•
Or, (to be proved on next slide) A
T
A=I.
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Math Modeling, I (GAUSS project)
2
Important Theorems
•
Theorem 1: If A
mxn
is an orthogonal matrix (m
≥
n), then
A
T
A=I
nxn
•
Proof. Let A = [
a
1
,
a
2
,…,
a
n
]. Since A is orthogonal,
a
i
T
a
j
=0
for all i
≠
j;
a
i
T
a
i
=1
•
Now, the (i,j) entry of A
T
A=
a
i
T
a
j
=
δ
ij.
•
So, A
T
A=I.
•
Theorem 2. A
T
A and AA
T
are symmetric matrices.
Important Theorems, cont’d
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 Spring '11
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