|
For sake of
clarity we do not write variables in italics in this module |
|
|
|
This is not a course in matrix
algebra (including vector and tensor calculus), but a quick reminder, assuming
you know the basic facts of life here. |
|
|
We also cut a lot of corners, not distinguishing
much between matrices (a mathematical object) and tensors (a physical object),
"true" vectors and "polar" vectors, Cartesian and
non-Cartesian coordinate systems and the like. |
|
|
We will deal with some topics of matrix algebra
roughly in the sequence they come up in the backbone chapters |
|
|
|
|
|
( |
a11 |
a12 |
a13 |
) |
a21 |
a22 |
a23 |
a31 |
a32 |
a33 |
|
|
|
A matrix then is an assembly of nine numbers arranged as shown
on the left. |
|
|
|
|
|
|
In a simplified way of speaking, a matrix (or better tensor)
allows to correlate vectors in a simple linear
way. |
|
|
Every component of the vector r =
(r1, r2, r3) can be expressed as a linear
function of all components of a second vector t = (t1,
t2, t3) by the equations |
|
|
|
|
|
r1 |
= |
a11 · t1 + a12 ·
t2 + a13 · t3 |
|
|
|
r2 |
= |
a21 · t1 + a22
· t2 + a23 · t3 |
|
|
|
r3 |
= |
a31 · t1 + a32
· t2 + a33 · t3 |
|
|
|
|
|
|
|
In matrix notation we simple write |
|
|
|
|
|
|
|
|
|
|
|
With A being the symbol for the matrix defined above. |
|
|
We then have already defined how a matrix is
multiplied with a vector and that a new vector is the result of the
multiplication. |
|
The matrix A, if interpreted as an entity that relates two vectors
with each other, must have certain
properties that are not required for a general matrix (that might
express, e.g., the coefficients of a linear system of equations with several
unknowns). |
|
|
If we change the coordinate system in which we
express the vectors, the components of the vectors will be different numbers,
but the vectors themselves (the arrows) stay unchanged. This imposes some
conditions on the set of nine numbers - the matrix - connecting the components
of the vectors and any matrix meeting these conditions
we call a tensor. |
|
|
A tensor thus
is a set of nine numbers, and the numerical value of these numbers depends on
the coordinate system in which the tensor is expressed. If we do a coordinate
transformation, the numerical values of the nine components must then transform
in a specific way. |
|
Transforming a coordinate system into another one
is done by matrices as follows: |
|
|
If the first vector r is chosen to
be one of the unit vectors defining some
Cartesian coordinate system, the second vector r' obtained by
multiplying r with the transformation matrix T, can be interpreted as the unit vector of some new coordinate system |
|
|
The set of unit vectors
ri with i = x,y,z will be changed to a new set
r'i by |
|
|
|
|
|
|
|
|
|
|
|
and T is called the transformation matrix. It is clear that
T must have certain properties if
the r'i are also supposed to be unit vectors |
|
|
While this is clear, it is not so clear what we
have to do if we want to reverse the transformation. The simple thing is to
write |
|
|
|
|
|
|
|
|
|
|
|
and defining ( T 1 ) to be the
inverse matrix to T so that the operation can be reversed. |
|
But how do we calculate the numerical values of
the components of T
1 if we know the numerical values of the components
of T ?? |
|
|
In order to be able to give a simple formula, we
first have to introduce something else, the determinant of a matrix |
|
|
|
|
The determinant |A| of a
matrix A is a single number calculated by summing up the diagonal
products in a special fashion. |
|
|
For a 3 × 3 matrix we have |
|
|
|
|
|
|A| |
= |
a11 · a22
· a33 + a12 · a23 ·
a31 + a13 · a21 ·
a32
a13 · a22 ·
a31 a11 · a23a ·
32 a12 · a21a ·
33 |
|
|
|
|
|
|
|
Look at the written matrix A above and you see that you start by
doing the products by going down diagonally from left to right, adding the
products of the three possible diagonals - always completing a diagonal by
repeating the matrix if necessary. Then you subtract the product you obtain by
going down the diagonal from right to left. |
|
|
This sounds more complicated as it is; graphically
it looks like this: |
|
|
|
|
|
|
|
|
|
|
The determinant of a matrix obtained
in this way is a number that comes up a lot in all kinds of matrix operation;
the same is true for a related quantity, the subdeterminant Aik of the
matrix A |
|
|
There are as many subdeterminants as
there are elements in the matrix. Aik is obtained by
- Erasing the line and the row that contains the element
aik and calculating the determinant of the matrix that
remains, and
- multiplying the number obtained by ( 1)i · k
|
|
With the concept of
a subdeterminant, we can also define the rank of
a matrix: |
|
|
The rank of a matrix is the number of row (or
columns, resp.) of the determinant or largest subdeterminant with non-zero
value. In other words, the rank of a 3 × 3 matrix A is rank(A) = 3 if |A|
¹ 0; if |A| = 0, you look for the
largest subdeterminant |
|
With determinant and
subdeterminant, the inverse matrix is easy to
formulate: |
|
|
The inverse matrix A 1 to A has the elements
(aik)1 given by |
|
|
|
|
|
|
|
|
|
|
|
i.e. the value of the respective
subdeterminant divided by the value of the determinant. Note that the indexes
are interchanged ("ik" →
"ki"); and that the " 1"
must be read as "inverse", it is
not an exponent!!! |
|
|
We will not prove it here; but it is
not too difficult - just solve the system of equations given above for the
ti. |
|
Two more important points follow
directly: |
|
|
An inverse matrix (
A1 ) to
A only exists if the determinant
of A is not
zero! |
|
|
The product of A1 and
A results in the
identity matrix I |
|
|
|
|
|
|
( |
1 |
0 |
0 |
) |
A1 ·
A = I = |
0 |
1 |
0 |
|
0 |
0 |
1 |
|
|
|
|
|
|
The last claim is unproved, we first
need the multiplication rule for matrices to prove it. |
|
|
Multiplication of the matrix
A with the matrix B gives a new matrix
C; and the element
cik of C is
obtained by taking the scalar product of the "line" or
"row" vector in row i of matrix
A times the column vector of
column k of matrix B. This
is best seen in a kind of graph: |
|
|
|
|
|
( |
× |
× |
× |
) |
|
( |
× |
× |
× |
) |
|
( |
× |
b12 |
× |
) |
× |
× |
× |
= |
× |
× |
× |
· |
× |
b22 |
× |
× |
c32 |
× |
|
a13 |
a12 |
a32 |
|
× |
b32 |
× |
|
|
|
|
|
|
|
Now it is still fairly messy, but straightforward
to prove the claim from above - you may want to try it. |
|
A useful relation is that the multiplication of
any matrix with the identity matrix I doesn't change anything. |
|
|
|
|
|
|
|
|
|
And this is also true for multiplying a vector
with I: |
|
|
|
|
|
|
|
|
|
|
From the various definitions you may get the
feeling, that signs are important and possibly tricky. Well, that's true. |
|
|
Matrix multiplication, in general is not
commutative, i.e. A ·
B ¹
B · A - you must watch out if you multiply from the left or
from the right. |
|
|
Still, we now can solve "mixed" vector -
matrix equations. Take, for example
|
|
|
|
|
|
|
|
|
|
|
|
Multiplying from the right with I yields. |
|
|
|
|
|
I ·
r0 = I
· A 1 ·
r0 + I ·
T(I) |
I ·
r0 = A
1 · r0 + I
· T(I) |
|
|
|
|
|
|
|
That looks a bit stupid, but with
this cheap trick we now we have only tensors in connection with
r0, which means we now can combine the
"factors" of r0, giving |
|
|
|
|
|
|
|
|
|
|
|
our O-lattice theory master
equation. |
|
One last important
property of transformation matrices is that their determinant gives directly
the volume ratio of the unit cells: |
|
|
|
|
|
|A| |
= |
V(after the transformation)
V(before the transformation) |
|
|
|
|
|
|
This is not particularly easy to see,
but simply consider two points: |
|
|
1. The base vector a(I) is
transformed to the base vector a(II) via |
|
|
|
|
|
|
|
|
|
|
|
2. The volumes V of elementary
cells is given by |
|
|
|
|
|
|
|
|
|
|
|
Since we produce the O-lattice from a
crystal lattice with the matrix I A
1, the volume VO of an
O-lattice cell (in units of the volume of a crystal unit cell) is |
|
|
|
|
|
|
|
|
|
|
|
Again, as remarked above; watch out for
signs. |
|
|
|
© H. Föll