Suppose that the point C is represented by the particular pair p1, p2 , and let D q1, q2 be any other point. Then we can find a formula for the length of CD. But under the transformation cf.
- Chapter 1.3.
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Thus we may speak of the roots p1, p2 , q1, q2 of f. It follows that the discriminant D0 of f0 must vanish as a consequence of the vanishing of D. We may also determine K by applying the transformation T to f and computing the explicit form of f0.
Therefore the discriminant of f is a relative invariant of T Lagrange ; and, in fact, the discriminant of f0 is always equal to the discriminant of f multiplied by the square of the determinant of the transformation.. Preliminary Geometrical Definition.
Invariance Principles and Conservation Laws
If there is associated with a geometric figure a quantity which is left unchanged by a set of transformations of the figure, then this quantity is called an absolute invariant of the set Halphen. Therefore the bilinear function h of the coefficients of two quadratic polynomials, representing the condition that their root pairs be harmonic conjugates, is a relative invariant of the transformation T. It is sometimes called a joint invariant, or simultaneous invariant of the two polynomials under the transformation.
Let the points u1, u2 v1, v2 , be harmonic conjugate to the pair p , r ; and also to the pair q , s.
In group theory , traces are known as " group characters. For square matrices and , it is true that. Lang , p. The trace is also invariant under a similarity transformation.
The trace of a product of two square matrices is independent of the order of the multiplication since. Therefore, the trace of the commutator of and is given by.
The trace of a product of three or more square matrices, on the other hand, is invariant only under cyclic permutations of the order of multiplication of the matrices, by a similar argument. The product of a symmetric and an antisymmetric matrix has zero trace,. The value of the trace for a nonsingular matrix can be found using the fact that the matrix can always be transformed to a coordinate system where the z -axis lies along the axis of rotation. In the new coordinate system which is assumed to also have been appropriately rescaled , the matrix is.
A Treatise on the Theory of Invariants
Lang, S. Linear Algebra, 3rd ed. New York: Springer-Verlag, pp. Munkres, J.