By Peter Woit
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Extra resources for Quantum Theory, Groups and Representations: An Introduction
Sample text
Once the first two rows are specified, the third row is uniquely determined. So as a manifold, SU (3) is eight dimensional, and one might think it could be identified with S 5 ×S 3 . It turns out that this is not the case, since the S 3 varies in a topologically non-trivial way as one varies the point in S 5 . As spaces, the SU (n) are topologically “twisted” products of odd dimensional spheres, providing some of the basic examples of quite non-trivial topological manifolds. 7 Eigenvalues and eigenvectors We have seen that the matrix for a linear transformation L of a vector space V changes by conjugation when we change our choice of basis of V .
Ln1 Ln2 . . Lnn vn We see from this that we can think of the transformed vector as we did above in terms of transformed coefficients vj with respect to fixed basis vectors, but also could leave the vj unchanged and transform the basis vectors. At times we will want to use matrix notation to write formulas for how the basis vectors transform in this way, and then will write e1 L11 L21 . . Ln1 e1 e2 L12 L22 . . Ln2 e2 .. → .. .. .. .
For an arbitrary n by n complex matrix there will be n solutions (counting repeated eigenvalues with multiplicity). A basis will exist for which the matrix will be in upper triangular form. The case of self-adjoint matrices L is much more constrained, since transposition relates matrix elements. 1 (Spectral theorem for self-adjoint matrices). Given a self-adjoint complex n by n matrix L, there exists a unitary matrix U such that U LU −1 = D where D is a diagonal matrix with entries Djj = λj , λj ∈ R.