CHECKLIST SECOND MIDTERM,
Math21b, O.Knill
The EIGENVECTORS AND EIGENVALUES of a matrix
A
reveal the structure of
A
.
Diagonalization in
general eases the computations with
A
. It allows to find explicit formulas for LINEAR DYNAMICAL SYSTEMS
x
7→
Ax
. Such systems are important for example in probability theory. The dot product leads to the notion of
ORTHOGONALITY, allows measurements of angles and lengths and leads to geometric notations like rotation,
reflection or projection in arbitrary dimensions. Least square solutions of
Ax
=
b
allow for example to solve
fitting problems. DETERMINANTS of matrices appear in the definition of the characteristic polyomial and as
volumes of parallelepipeds or as scaling values in change of variables. The notion allows to give explicit formulas
for the inverse of a matrix or to solutions of
Ax
=
b
.
ORTHOGONAL
~v
·
~w
= 0.
LENGTH

~v

=
√
~v
·
~v
.
UNIT VECTOR
~v
with

~v

=
√
~v
·
~v
= 1.
ORTHOGONAL SET
v
1
, . . . , v
n
: pairwise orthogonal.
ORTHONORMAL SET
orthogonal and length 1.
ORTHONORMAL BASIS
A basis which is orthonormal.
ORTHOGONAL TO V
v
is orthogonal to
V
if
v
·
x
= 0 for all
x
∈
V
.
ORTHOGONAL COMPLEMENT OF V
Linear space
V
⊥
=
{
v

v
orthogonal to
V
}
.
PROJECTION ONTO V
orth. basis
v
1
, . . . , v
n
in
V
, perp
V
(
x
) = (
v
1
·
x
)
v
1
+
. . .
+ (
v
n
·
x
)
v
n
.
GRAMMSCHMIDT
Recursive
u
i
=
v
i

proj
V
i

1
v
i
,
w
i
=
u
i
/

u
i

leads to orthonormal basis.
QRFACTORIZATION
Q
= [
w
1
· · ·
w
n
],
R
ii
=
u
i
, [
R
]
ij
=
w
i
·
v
j
, j > i
.
TRANSPOSE
[
A
T
]
ij
=
A
ji
. Transposition switches rows and columns.
SYMMETRIC
A
T
=
A
.
SKEWSYMMETRIC
A
T
=

A
(
⇒
R
=
e
A
orthogonal:
R
T
=
e
A
T
=
e

A
=
R

1
).
DOT PRODUCT AS MATRIX PRODUCT
v
·
w
=
v
T
·
w
.
ORTHOGONAL MATRIX
Q
T
Q
= 1.
ORTHOGONAL PROJECTION
onto
V
is
AA
T
, colums
~v
i
are orthonormal basis in
V
.
ORTHOGONAL PROJECTION
onto
V
is
A
(
A
T
A
)

1
A
T
, columns
~v
i
are basis in
V
.
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 Spring '03
 JUDSON
 Linear Algebra, Algebra, Eigenvectors, Vectors, Formulas, Det

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