Editing Orthogonal matrix (section)

===Matrix properties===
A real square matrix is orthogonal [[if and only if]] its columns form an [[orthonormal basis]] of the [[Euclidean space]] {{math|'''R'''<sup>''n''</sup>}} with the ordinary Euclidean [[dot product]], which is the case if and only if its rows form an orthonormal basis of {{math|'''R'''<sup>''n''</sup>}}. It might be tempting to suppose a matrix with orthogonal (not orthonormal) columns would be called an orthogonal matrix, but such matrices have no special interest and no special name; they only satisfy {{math|1=''M''<sup>T</sup>''M'' = ''D''}}, with {{mvar|D}} a [[diagonal matrix]].

The [[determinant]] of any orthogonal matrix is +1 or −1. This follows from basic facts about determinants, as follows:
<math display="block">1=\det(I)=\det\left(Q^\mathrm{T}Q\right)=\det\left(Q^\mathrm{T}\right)\det(Q)=\bigl(\det(Q)\bigr)^2 .</math>

The converse is not true; having a determinant of ±1 is no guarantee of orthogonality, even with orthogonal columns, as shown by the following counterexample.
<math display="block">\begin{bmatrix}
2 & 0 \\
0 & \frac{1}{2}
\end{bmatrix}</math>

With permutation matrices the determinant matches the [[even and odd permutations|signature]], being +1 or −1 as the parity of the permutation is even or odd, for the determinant is an alternating function of the rows.

Stronger than the determinant restriction is the fact that an orthogonal matrix can always be [[diagonalizable matrix|diagonalized]] over the [[complex number]]s to exhibit a full set of [[Eigenvalues and eigenvectors|eigenvalues]], all of which must have (complex) [[absolute value|modulus]]&nbsp;1.