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Mathematics-Online course: Linear Algebra - Matrices - Special Matrices

Isometry of Orthogonal and Unitary Matrices

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A real (complex) matrix $ A$ is orthogonal (unitary) if and only if the Euclidian vector norm is invariant under $ A$, that is,

$\displaystyle \vert Av\vert = \vert v\vert\, . $

(Authors: App/Burkhardt/Höllig)

Only the complex case, which includes the real one, has to be considered.

(i)
If $ A$ is unitary, then

$\displaystyle \langle A x, Ay \rangle = \langle A^\ast A x, y \rangle = \langle x,y \rangle\,. $

Thus it follows the invariance of the complex scalar product, which includes the norm invariance.

(ii)
From the norm invariance it follows that the columns $ v_j$ of $ A$ have norm 1, since they are images of the unit vectors. To prove that different columns are orthogonal choose $ \lambda = \exp(\mathrm{i}\vartheta)$, such that

$\displaystyle \langle v_j,\lambda v_k\rangle \in \mathbb{R}\, . $

Then by the norm invariance and by the definition of norm we obtain

$\displaystyle 2 = \vert e_j+\lambda e_k\vert^2 = \vert v_j+\lambda v_k\vert^2 ... ...e v_j,\lambda v_k\rangle = 2+2 \overline{\lambda}\langle v_j,v_k\rangle \,, $

hence the scalar product on the right hand side vanishes.

(Authors: App/Burkhardt/Höllig)
  automatically generated 4/21/2005