Exercises

Apr 22'25

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Prove that the symmetry and projection with respect to the [math]Ox[/math] axis rotated by an angle [math]t/2\in\mathbb R[/math] are given by the matrices

[[math]] S_t=\begin{pmatrix}\cos t&\sin t\\ \sin t&-\cos t\end{pmatrix} [[/math]]

[[math]] P_t=\frac{1}{2}\begin{pmatrix}1+\cos t&\sin t\\ \sin t&1-\cos t\end{pmatrix} [[/math]]

and then diagonalize these matrices, and if possible without computations.

BBot

Apr 22'25

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Prove that the isometries of [math]\mathbb R^2[/math] are rotations or symmetries,

[[math]] R_t=\begin{pmatrix}\cos t&-\sin t\\ \sin t&\cos t\end{pmatrix}\quad,\quad S_t=\begin{pmatrix}\cos t&\sin t\\ \sin t&-\cos t\end{pmatrix} [[/math]]

and then try as well to find a formula for the isometries of [math]\mathbb R^3[/math].

BBot

Apr 22'25

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Prove that the isometries of [math]\mathbb C^2[/math] of determinant [math]1[/math] are

[[math]] U=\begin{pmatrix}a&b\\ -\bar{b}&\bar{a}\end{pmatrix}\quad,\quad |a|^2+|b|^2=1 [[/math]]

then work out as well the general case, of arbitrary determinant.

BBot

Apr 22'25

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Prove that the flat matrix, which is the all-one [math]N\times N[/math] matrix, diagonalizes over the complex numbers as follows,

[[math]] \begin{pmatrix} 1&\ldots&\ldots&1\\ \vdots&&&\vdots\\ \vdots&&&\vdots\\ 1&\ldots&\ldots&1\end{pmatrix}=\frac{1}{N}\,F_N \begin{pmatrix} N\\ &0\\ &&\ddots\\ &&&0\end{pmatrix}F_N^* [[/math]]

where [math]F_N=(w^{ij})_{ij}[/math] with [math]w=e^{2\pi i/N}[/math] is the Fourier matrix, with the convention that the indices are taken to be [math]i,j=0,1,\ldots,N-1[/math].