450 Appendix
a vector u can be formed from the basis vectors and components
u = u 1 ê 1 + u 2 ê 2 + u 3 ê 3 + u 4 ê 4 + u 5 ê 5 = (u 1 , u 2 , u 3 , u 4 , u 5 ).
(45)
This vector is defined in a five-dimensional space, with five axes
each orthogonal to the others, because their scalar products are
zero. Although this is difficult to visualize (or draw), the mathematics carries through directly from the three-dimensional
case. N mutually orthogonal vectors thus provide a basis for an
N-dimensional space.
These ideas are formalized in terms of vectors in a general
linear vector space. For our purposes, a vector space is a collection of vectors x, y, z, satisfying several criteria:
• The sum of any two vectors in the space is also in the
space.
• Vector addition commutes: x + y = y + x.
• Vector addition is associative: (x + y) + z = x + (y + z).
• There exists a unique vector 0 such that for all x, x = x + 0.
• There exists a unique vector −x such that for all x, x + (−x)
= 0.
• Scalar multiplication is associative: α(β x) = (αβ)x.
• Scalar multiplication is distributive: α(x + y) = α x + αy
and (α + β)x = (α x + βx).
A point worth considering is the number of independent
vectors in a vector space. Given N vectors x 1 , x 2 , . . . , x N in a
linear vector space, a weighted sum ∑α i x i is called a linear
combination. The N vectors are linearly independent if
i
N
=
∑
1
α i x i = 0 only when all α i = 0,
(46)
so that no vector can be expressed as a combination of the
others. Otherwise, the vectors are linearly dependent, and one
can be expressed as a linear combination of the others.
This idea corresponds to that of basis vectors. If N basis
vectors are mutually orthogonal, they are linearly independent.
Because any vector in an N-dimensional space is a linear combination of N linearly independent basis vectors, the basis
vectors span the space. Thus the dimension of a vector space
is the number of linearly independent vectors within it. For
example, we cannot find four linearly independent vectors in
three dimensions.
Though vector spaces sound abstract, they are useful in
seismology. For example, in Chapter 2 we represent travelling
waves by normal modes, which are orthogonal basis vectors in
a vector space, so any wave is a weighted sum of them. The
modes of a string (Section 2.2.5) form a Fourier series (Chapter 6), in which a function is expanded into sine and cosine
functions that are the basis vectors of a vector space. A similar approach is also used for the modes of the spherical earth
(Section 2.9). Vector space ideas are also used in inverting
seismological observations to study earth structure (Chapter
7).
A.4 Matrix algebra
A.4.1 Definitions
Matrix algebra is a powerful tool often used to study systems of
equations. As a result, it appears in seismological applications,
including stresses and strains, locating earthquakes, and seismic
tomography. We thus review some basic ideas, often stating
results without proof and leaving proofs for the problems. Further discussion of these topics can be found in linear algebra texts.
Given a matrix A with m rows and n columns, called an
m × n matrix,
A
a
a
a
a
a
a
a
a
a
n
n
m
m
mn
. . .
. . .
.
.
. . .
.
.
.
. . .
.
. . .
=
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
11
12
1
21
22
2
1
2
(1)
and a second matrix B, also with m rows and n columns,
matrix addition is defined by
A B
a
b
a
b
a
b
a
b
a
b
a
b
a
b
a
b
a
b
n
n
n
n
m
m
m
m
m n
m n
. . .
. . .
.
.
. . .
.
.
.
. . .
.
. . .
+ =
+
+
+
+
+
+
+
+
+
⎛
⎝
⎜
⎜
⎜
⎜
11
11
12
12
1
1
21
21
22
22
2
2
1
1
2
2
⎜ ⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
.
(2)
The usual convention is to indicate matrices with capital letters
and their elements with lower-case ones.
Matrix multiplication is defined such that for a matrix A that
is m × n and a matrix B that is n × r, the ij th element of the m × r
product matrix C = AB is defined by
c ij =
k
n
=
∑
1
a ik b kj = a ik b kj .
(3)
The ij th element of C is the scalar product of the i th row of A
and the j th column of B. As a result, for matrix multiplication
the two matrices need not have the same number of rows
and columns, but must have the number of columns in the first
matrix equal to the number of rows in the second. Often the
numbers of rows and columns in the two matrices allow multiplication in only one order. Thus, in the example above, A
“premultiplies” B, or B “postmultiplies” A. A convenient way
to remember this is that the number of columns in the first
matrix must equal the number of rows in the second, but this
dimension does not appear in the product. In the case of AB =
C, written schematically, we have [m × n][n × r] = [m × r].
Hence, in the final form in Eqn 3, the summation convention
shows that k is summed out, leaving i and j as free indices, so c ij
is a matrix element. Furthermore, even if both AB and BA are
allowed, the two products are generally not equal, so matrix
multiplication is not commutative.
a vector u can be formed from the basis vectors and components
u = u 1 ê 1 + u 2 ê 2 + u 3 ê 3 + u 4 ê 4 + u 5 ê 5 = (u 1 , u 2 , u 3 , u 4 , u 5 ).
(45)
This vector is defined in a five-dimensional space, with five axes
each orthogonal to the others, because their scalar products are
zero. Although this is difficult to visualize (or draw), the mathematics carries through directly from the three-dimensional
case. N mutually orthogonal vectors thus provide a basis for an
N-dimensional space.
These ideas are formalized in terms of vectors in a general
linear vector space. For our purposes, a vector space is a collection of vectors x, y, z, satisfying several criteria:
• The sum of any two vectors in the space is also in the
space.
• Vector addition commutes: x + y = y + x.
• Vector addition is associative: (x + y) + z = x + (y + z).
• There exists a unique vector 0 such that for all x, x = x + 0.
• There exists a unique vector −x such that for all x, x + (−x)
= 0.
• Scalar multiplication is associative: α(β x) = (αβ)x.
• Scalar multiplication is distributive: α(x + y) = α x + αy
and (α + β)x = (α x + βx).
A point worth considering is the number of independent
vectors in a vector space. Given N vectors x 1 , x 2 , . . . , x N in a
linear vector space, a weighted sum ∑α i x i is called a linear
combination. The N vectors are linearly independent if
i
N
=
∑
1
α i x i = 0 only when all α i = 0,
(46)
so that no vector can be expressed as a combination of the
others. Otherwise, the vectors are linearly dependent, and one
can be expressed as a linear combination of the others.
This idea corresponds to that of basis vectors. If N basis
vectors are mutually orthogonal, they are linearly independent.
Because any vector in an N-dimensional space is a linear combination of N linearly independent basis vectors, the basis
vectors span the space. Thus the dimension of a vector space
is the number of linearly independent vectors within it. For
example, we cannot find four linearly independent vectors in
three dimensions.
Though vector spaces sound abstract, they are useful in
seismology. For example, in Chapter 2 we represent travelling
waves by normal modes, which are orthogonal basis vectors in
a vector space, so any wave is a weighted sum of them. The
modes of a string (Section 2.2.5) form a Fourier series (Chapter 6), in which a function is expanded into sine and cosine
functions that are the basis vectors of a vector space. A similar approach is also used for the modes of the spherical earth
(Section 2.9). Vector space ideas are also used in inverting
seismological observations to study earth structure (Chapter
7).
A.4 Matrix algebra
A.4.1 Definitions
Matrix algebra is a powerful tool often used to study systems of
equations. As a result, it appears in seismological applications,
including stresses and strains, locating earthquakes, and seismic
tomography. We thus review some basic ideas, often stating
results without proof and leaving proofs for the problems. Further discussion of these topics can be found in linear algebra texts.
Given a matrix A with m rows and n columns, called an
m × n matrix,
A
a
a
a
a
a
a
a
a
a
n
n
m
m
mn
. . .
. . .
.
.
. . .
.
.
.
. . .
.
. . .
=
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
11
12
1
21
22
2
1
2
(1)
and a second matrix B, also with m rows and n columns,
matrix addition is defined by
A B
a
b
a
b
a
b
a
b
a
b
a
b
a
b
a
b
a
b
n
n
n
n
m
m
m
m
m n
m n
. . .
. . .
.
.
. . .
.
.
.
. . .
.
. . .
+ =
+
+
+
+
+
+
+
+
+
⎛
⎝
⎜
⎜
⎜
⎜
11
11
12
12
1
1
21
21
22
22
2
2
1
1
2
2
⎜ ⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
.
(2)
The usual convention is to indicate matrices with capital letters
and their elements with lower-case ones.
Matrix multiplication is defined such that for a matrix A that
is m × n and a matrix B that is n × r, the ij th element of the m × r
product matrix C = AB is defined by
c ij =
k
n
=
∑
1
a ik b kj = a ik b kj .
(3)
The ij th element of C is the scalar product of the i th row of A
and the j th column of B. As a result, for matrix multiplication
the two matrices need not have the same number of rows
and columns, but must have the number of columns in the first
matrix equal to the number of rows in the second. Often the
numbers of rows and columns in the two matrices allow multiplication in only one order. Thus, in the example above, A
“premultiplies” B, or B “postmultiplies” A. A convenient way
to remember this is that the number of columns in the first
matrix must equal the number of rows in the second, but this
dimension does not appear in the product. In the case of AB =
C, written schematically, we have [m × n][n × r] = [m × r].
Hence, in the final form in Eqn 3, the summation convention
shows that k is summed out, leaving i and j as free indices, so c ij
is a matrix element. Furthermore, even if both AB and BA are
allowed, the two products are generally not equal, so matrix
multiplication is not commutative.