Matrices:

The rotation matrices are only one example of quantities with two indices which are called matrices by mathematicians.

It is possible to define calculation rules generally for matrices, i.e. schemes with lines and columns and to examine their structures. We want to restrict our considerations to quadratic -matrices and even more specifically to -matrices with real elements.

We denote the matrices by underlined capital letters, e.g. . Their elements carry two indices: the left one denotes the (horizontal) line and the right one the (vertical) column of the matrix:

matrix:

Some kinds of matrices have special names because of their importance:

In particular, diagonal matrices are of special importance, having only the three elements , and along the so-called main diagonal (:from the left on top downward to the right) different from . The second diagonal (:from the right up downward to the left) is, in comparison, much less important.

diagonal matrix:

The matrices of rotations by a multiple of are examples of diagonal matrices: , and .

Half the way to the diagonal structure the triangle form is worth mentioning, which has only zeros either above or below the main diagonal:

triangle matrix:

Also matrices in  box form are especially convenient for many purposes. In these matrices non-zero elements are only in "boxes" around the main diagonal. Our rotation matrices and are of this kind.

matrix in  box form:

A simple operation that can be carried out with every matrix is transposition: This means the reflection of all matrix elements through the main diagonal, or in other words, the exchange of lines and columns:

transposed matrix:

There exist matrices for which transposition does not change anything: They are called symmetric.

symmetric matrix:

These symmetric matrices occur very often in physics and have the advantage that they can be brought to diagonal form by certain simple transformations.

As you see immediately, each symmetric matrix has only six independent elements.

If the reflection through the main diagonal leads to a minus sign, the matrix is called antisymmetric:

antisymmetric matrix:

Of course the diagonal elements have to vanish in this case. Apparently an antisymmetric -matrix has only three independent elements. That is the deeper reason for the existence of a vector product in three dimensions, as we will soon see in more detail.

Finally we mention a special quantity of every matrix: The sum of the elements along the main diagonal  is called the  trace (:in German "Spur") of the matrix:

trace:

Much more important for physics is however the  multiplication of two -matrices  which corresponds in the case of transformation matrices to two transformations of the coordinate system carried out one after the other:

The following multiplicative instruction holds:

In the last part above the summation symbol is omitted according to the Einstein summation convention (s. Section 9.2.3.2), since the two identical indices signalize the summation well enough.

To calculate the product matrix element in the z-th line and the s-th column you may imagine the s-th (vertical) column of the factor matrix on the right side put horizontally upon the z-th line of the left factor matrix , elements on top of each other multiplied and the three products added: e.g. , thus altogether:

Exercise 9.10: Matrix multiplication Multiply the following transformation matrices: a)      and compare with Solution b)      especially to be compared with Solution c)      and compare with Solution d)      and compare with . Solution

The most important discovery to be made by working throegh the Exercise 9.10 is the fact that generally no commutative law  holds for rotations, and consequently not for the representing matrices. You can easily check this visually with every match box as is illustrated in the following Figure:

The examples from Exercise 9.10 have already shown to you that in some  exceptional cases the commutative law  nevertheless holds: all rotations around one and the same axis are for instance commutable. Also all diagonal matrices are commutable with each other. This is the reason for their popularity. If , the so-called commutation relation promises to be an interesting quantity. This will acquire great significance in quantum mechanics later on.

Apart from commutability, matrix multiplication behaves as expected: There holds an

Associative Law : .

Exercise 9.11: Associative Law for matrix multiplication Verify the Associative Law for the Euler rotation: which leads us from the space fixed coordinate system to the body fixed system of a rotating gyroscope. Solution

A uniquely determined

unit matrix: with ,

exists independently whether you multiply from the right or the left.

Only with the

inverse matrix: with

do we encounter a certain complication analogous to the condition for division by a real number. A uniquely determined inverse matrix exists only for the so-called  non-singular  matrices. These are matrices whose  determinant  does not vanish: . The determinants, the most important characteristics of matrices, will be treated in an extra insert in the following subsection.

For our transformation matrices, however, this constraint is unimportant. For these matrices the inverse is simply the transposed one which exists in every case as we have seen: Mathematicians call these matrices orthogonal (s. Section 9.8.2) and we will inspect these carefully later on:

orthogonal matrix: or

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