in the sense that any product of these elements is equal to a numerical multiple of a third one. In terms of this fundamental system we define the angle between any two spaces. Each of the complexes of the fundamental system is an infinite locus for spaces complimentary to it. The entire system is invariant under a group of collineations of the same order as the Euclidean group of motions. Degenerate cases are obtained by taking sections having a special relation to the fundamental system.

MATRICES IN THREE DIMENSIONS.

2. Progressive Matrices. We represent a point A in three dimensions by a set of four homogeneous coordinates a;. These coordinates a¡. determine a matrix

which may be used to represent the point. Two matrices of this kind will be called equal when their corresponding elements are equal. The matrix is zero if all its elements are zero. If a; =

write

A = k B.

k b, we shall

In this case the matrices A and B represent the same point but with different magnitudes. A linear function of A and B is defined by the matrix

In a similar manner we define any linear function of points or matrices A, B, C, etc. If the result does not vanish it represents a point in the space determined by A, B, C, etc. If it vanishes and the coefficients are not all zero those points lie in a lower space than a like number of points usually determine.

The coordinates of the line joining A and B are proportional to the two-rowed determinants in the matrix

We shall call the elements of this matrix the two-rowed determinants

ai ak b1 b

[This is not in conformity with the usual definition which makes element equivalent to coordinate a, or b; but is the only definition

which has a value in the present discussion.] The matrix is zero when all its elements are zero. In that case the points A and B have proportional coordinates and hence coincide. If the matrix is not zero it represents the line A B in the sense that from the matrix can be obtained the coordinates of the line. Conversely if the line is given a matrix can be formed by taking any two points on the line. Different matrices representing the same line are multiples of any For if A, B and P, Q are pairs of distinct points on the line A + 1⁄2 B, Q = με Α + με Β,

one.

and

P =

Thus a two-rowed matrix in addition to representing a line, has a definite size.

The matrix [A B] is in reality a set of six determinants

The sum of two matrices [A B] and [C D] is then a complex matrix, each element of which is the sum of corresponding elements in [A B] and [CD]. In general this sum cannot be represented as a single two-rowed matrix, just as the sum of corresponding Plücker coordinates of two lines are not in general coordinates of a line. For analytical purposes we express this sum by simply writing the two matrices with an addition sign between them. If, however, the lines A B and C D intersect in a point P, we can find points Q and R on those lines such that

Then

[A B] = [P Q],
[CD] = [PR].

We can consider [A B] as a product of A and B. For

[A(B+C)] = [A B] + [A C]

as we have just seen in the case of [P(Q + R)]. The process of multiplication consists in placing the second matrix under the matrix A

to form a two rowed matrix [A B]. We shall call this the progressive product. From the definition it is evident that

3. If the points A, B, C are not collinear, the coordinates of the plane ABC are proportional to the three rowed determinants in the matrix

This matrix represents the plane in the sense that from it we can determine the coordinates of the plane. Conversely to represent any plane as a matrix we take three non-collinear points of the plane and form the matrix from them. The elements of such a matrix are the three-rowed determinants belonging to it. In reality we are considering this as a one-rowed matrix of four terms (equal to the three-rowed determinants in [A B C]) arranged in some definite order. Two matrices of this kind will be called equal if corresponding elements are equal and are added by adding corresponding elements.

If P, Q, R are any three points of the plane determined by A, B, C, PA+ B+ λz C,

Thus a matrix [P Q R] in addition to representing a plane has a definite size. The vanishing of [P Q R] signifies that P, Q, R lie on a line.

The matrix [A B C] can be regarded as a product of [A B] and C, A and [B C] or of A, B and C, the process of multiplication consisting always of placing the first matrix at the top and the others in order under it to form a single matrix.2

2 In this multiplication each matrix must have four columns. If instead of [A B] we have a complex the operation must be performed distributively on each two-rowed matrix of the sum. For purposes of addition we regard our quantities as matrices of one row but for purposes of multiplication as matrices or sums of matrices of four columns.

The sum of any number of three-rowed matrices can be expressed as a single three-rowed matrix [P Q R]. In fact let A1 B1 C1 and A2 B2 C2 cut in a line PQ. Then

From four points A, B, C, D we can form a four rowed matrix or determinant

This matrix has only one element and hence we write it as a determinant. A matrix of one element is analytically equivalent to a number. We use the parentheses to indicate this fact. Square brackets are used to represent matrices which do not reduce to numbers. The vanishing of (A B C D) is the condition that the four points lie in a plane.

The quantity (A B C D) can be regarded as a product in a number of ways. From the definition it is evident that

(ABCD)

=

(A · B C D) = (A B·C D) = — (A B D C).

4. Regressive Matrices. We can consider space as generated by planes as well as by points. If its coordinates are a, a plane a is then represented by a matrix

The same plane may be represented by a matrix [A B C]. Then the

coordinates a are proportional to the coefficients of x in the determinant A B C X|.3

If a, is equal to the coefficients of x; in that determinant we shall write

α= = [A B C].

Thus a three-rowed matrix is for our purpose equivalent to a one-rowed matrix in contragredient variables.

The line of intersection of two planes a and ẞ can be represented by a matrix

[a b] =
B]

а1 а2 аз 04

B1 B2 B3 B4

If the same line is the join of two points A and B we know from analytical geometry that the coordinates qik are proportional to the coefficients of the minors in the determinant | A B X Y. If it is equal to the coefficient of x; y in the determinant we shall write

Yi Yk

[a ß] = [A B].

This amounts to saying that in the determinant [A B a ßl, each minor in the first two rows is then equal to its algebraic compliment (coefficient in the expansion of the determinant).

Similarly we represent the point of intersection of three planes by a matrix [ay]. The coordinates a; of this point A are proportional to the coefficients of in the determinant [ay]. In particular if a, is equal to the coefficient of §; in that determinant we write

Α = [αβ γ].
A

In this case each term of the first row in the determinant (A a B Y) is equal to its algebraic compliment.

There is a determinant [a ß y ] of four planes just as of four points. These quantities [a B], [a ẞy], (a ẞy d) can be regarded as products formed according to the same laws as the products of points. These products of matrices expressed in plane coordinates we shall call regressive.

3 It is to be observed that here X is written last. If we take the coefficients of X; in the determinant X A B C❘ they will have different signs from the coefficients used here.