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Selected Properties of Matrices,
Determinants, Solutions of Linear
Equations, Eigenvalues, and Eigenvectors
Last Updated: October 24, 2007
In what follows,
A
is an n by n real matrix unless otherwise stated.
The inverse of
A
is denoted by
A
1
and its determinant is 
A
.
Column vectors are denoted by lower case bold letters; e.g.
x
,
y
,
and
z
and can be assumed to be of length n unless otherwise stated.
The transpose of a matrix is
A
T
and a row vector is
x
T
.
1.0 Solution of Linear Equations
1.1
Equivalent Statements Regarding
Ax
=
b
a.
A
is nonsingular
b.
Ax
=
b
has a unique nonzero solution
c.
Ax
=
0
has only
x
=
0
as its unique solution
d.

A

≠
0
e.
A
1
exists and
A
1
A
=
AA
1
=
I
(the identity matrix)
f.
Gaussian elimination works (with row swaps when
necessary to avoid a zero pivot).
g.
Row operations on
A
do not produce a zero row
1.2
Matrix Decompositions
a.
A
=
L
U
=
LU
=
L
DU
if the upperleft principal
submatrices are nonsingular where
L
is unit lower,
U
is unit upper and
D
is diagonal.
b.
If
A
is symmetric and positive definite, then
A
=
L
D
1/2
D
1/2
U
=
LL
T
where
D
1/2
is the diagonal matrix
whose entries are the square root of the diagonals
of
D
and
L
=
L
D
1/2
= (
D
1/2
U
)
T
1.3. Solving
Ax
=
b
by Iteration (Jacobi (JA), GaussSeidel
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 Winter '10
 Taylor

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