Geoscience Reference
In-Depth Information
Matrix transpose:
A
ij
=
A
ji
.
(A
+
B)
=
A
+
B
.
Matrix multiplication: An
(n
×
m)
matrix
A
times an
(m
×
p)
matrix
B
produces an
(n
×
p)
matrix
C
, with
C
ij
=
k
=
1
A
ik
B
kj
.
(AB)C
=
A(BC)
,
A(B
+
C)
=
AB
+
AC
,
(A
+
B)C
=
AC
+
BC
,
(AB)
=
B
A
. Note in general
AB
=
BA
.
The following properties apply to square matrices.
=
0
,
∀
i
=
j
. The identity matrix
I
is diagonal with
I
ii
=
1.
AI
=
IA
=
A
Diagonal matrix:
A
ij
for all square
A
.
Some
square matrices have inverses:
AA
−
1
=
A
−
1
A
=
I
.
(AB)
−
1
=
B
−
1
A
−
1
.
(A
)
−
1
=
(A
−
1
)
.
The trace is the sum of diagonal elements (or eigenvalues): Tr
(A)
=
i
A
ii
.
|
aA
|=
a
n
The determinant
|
A
|
is the product of eigenvalues.
|
AB
|=|
A
||
B
|
,
|
a
|=
a
,
|
A
|
,
A
−
1
|
|=
1
/
|
A
|
. A matrix
A
is invertible iff
|
A
| =
0.
If
n
square matrix
A
,
A
is said to be singular. This means at least one
column is linearly dependent on (i.e., a linear combination of ) other columns (same for rows). Once
all such linearly dependent columns and rows are removed,
A
is reduced to a smaller
r
×
r
matrix,
and
r
is called the rank of
A
.
|
A
|=
0 for an
n
×
An
n
×
n
matrix
A
has
n
eigenvalues
λ
i
and eigenvectors (up to scaling)
u
i
, such that
Au
i
=
λ
i
u
i
.
In general, the
λ
's are complex numbers. If
A
is real and symmetric,
λ
's are real numbers, and
u
's are
orthogonal. The
u
's can be scaled to orthonormal, i.e., length one, so that
u
i
u
j
=
I
ij
. The spectral
decomposition is
A
=
i
λ
i
u
i
u
i
=
i
. For invertible
A
,
A
−
1
λ
i
u
i
u
i
1
.
A real symmetric matrix
A
is positive semi-definite if its eigenvalues
λ
i
≥
0. An equivalent
n
,x
Ax
≥
0. It is strictly positive definite if
λ
i
>
0 for all
i
.
condition is
∀
x
∈ R
A positive semi-definite matrix has rank
r
equal to the number of positive eigenvalues. The
remaining
n
−
r
eigenvalues are zero.
n
, we have
For a vector
x
∈ R
0-norm:
x
0
=
count of nonzero elements
1
=
i
=
1
|
1-norm:
x
i
|
2-norm (the Euclidean norm, the length, or just “the norm”):
x
x
2
=
i
=
1
x
i
1
/
2
max
i
=
1
|
∞
-norm:
x
∞
=
x
i
|
