16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 10.4.14

Bilinear Forms

Mathematics — Linear Algebra · lorentz-type-operator-c2

Problem

Let

T

be a linear operator on

C^2

which preserves the quadratic form

x_1^2 - x_2^2

. Show that (a)

\det(T) = \pm 1

. (b) If

M

is the matrix of

T

in the standard basis, then

M_{22} = \pm M_{11}

M_{21} = \pm M_{12}

M_{11}^2 - M_{12}^2 = 1

. (c) If

\det M = 1

, then there is a non-zero complex number

c

such that

M = \frac{1}{2}\begin{bmatrix} c + \dfrac{1}{c} & c - \dfrac{1}{c} \\[6pt] c - \dfrac{1}{c} & c + \dfrac{1}{c} \end{bmatrix}.

(d) If

\det M = -1

then there is a complex number

c

such that

M = \frac{1}{2}\begin{bmatrix} c + \dfrac{1}{c} & c - \dfrac{1}{c} \\[6pt] -c + \dfrac{1}{c} & -c - \dfrac{1}{c} \end{bmatrix}.