Kunze Linear Algebra · Problem 3.3.3

Linear Transformations

Mathematics — Methods · isomorphism-hermitian-matrices

Problem

Let

W

be the set of all

2 \times 2

complex Hermitian matrices, that is, the set of

2 \times 2

complex matrices

A

such that

A_{ij} = \overline{A_{ji}}

(the bar denoting complex conjugation). As we pointed out in Example 6 of Chapter 2,

W

is a vector space over the field of real numbers, under the usual operations. Verify that

(x, y, z, t) \to \begin{bmatrix} t + x & y + iz \\ y - iz & t - x \end{bmatrix}

is an isomorphism of

R^4

onto

W