# Linear algebra/Orthogonal matrix

Alternative notations
${\displaystyle Q^{-1}=Q^{\mathrm {T} }}$ ${\displaystyle Q^{\mathrm {T} }Q^{-1}=Q^{-1}Q^{\mathrm {T} }=I}$
${\displaystyle {\underline {\underline {Q}}}^{-1}={\underline {\underline {Q}}}^{\mathrm {T} }}$ ${\displaystyle {\underline {\underline {Q}}}^{\mathrm {T} }\cdot {\underline {\underline {Q}}}^{-1}={\underline {\underline {Q}}}^{-1}\cdot {\underline {\underline {Q}}}^{\mathrm {T} }={\underline {\underline {I}}}}$
${\displaystyle Q_{jk}^{-1}=Q_{kj}\equiv Q_{jk}^{\mathrm {T} }}$ ${\displaystyle \sum _{k}Q_{ki}Q_{kj}^{-1}=\sum _{k}Q_{ik}^{-1}Q_{jk}=\delta _{ij}}$

A real-valued and square matrix ${\displaystyle Q}$ is orthogonal (or orthonormal) if ${\displaystyle Q^{\mathrm {T} }=Q^{-1},}$ where ${\displaystyle Q^{\mathrm {T} }}$ is the transpose and ${\displaystyle Q^{-1}}$ is its inverse. Equivalently, ${\displaystyle Q}$ is orthogonal when ${\displaystyle Q^{\mathrm {T} }Q=QQ^{\mathrm {T} }=I,}$ where ${\displaystyle I}$ is the identity matrix. It has a number of well-known properties:[1]

### Example

Since orthogonal matrices form a group under multiplication, we can construct a non-trivial orthogonal matrix my multiplying two matrices that are easy to understand. Consider, for example, ${\displaystyle {\underline {\underline {C}}}={\underline {\underline {A}}}\cdot {\underline {\underline {B}}}:}$

${\displaystyle \overbrace {\begin{bmatrix}{\frac {1}{\sqrt {2}}}&{\frac {1}{\sqrt {2}}}&0\\-{\frac {1}{\sqrt {2}}}&{\frac {1}{\sqrt {2}}}&0\\0&0&-1\end{bmatrix}} ^{\underline {\underline {A}}}\cdot \overbrace {\begin{bmatrix}1&0&0\\0&{\frac {1}{\sqrt {2}}}&{\frac {1}{\sqrt {2}}}\\0&-{\frac {1}{\sqrt {2}}}&{\frac {1}{\sqrt {2}}}\end{bmatrix}} ^{\underline {\underline {B}}}=\overbrace {\begin{bmatrix}{\frac {1}{\sqrt {2}}}&{\frac {1}{2}}&{\frac {1}{2}}\\-{\frac {1}{\sqrt {2}}}&{\frac {1}{2}}&{\frac {1}{2}}\\0&{\frac {1}{\sqrt {2}}}&-{\frac {1}{\sqrt {2}}}\\\end{bmatrix}} ^{\underline {\underline {C}}}}$

Both ${\displaystyle {\underline {\underline {A}}}}$ and ${\displaystyle {\underline {\underline {B}}}}$ posses a symmetry that could lead one to postulate a non-existent symmetry among off-diagonal elements. But no such symmetry exists for ${\displaystyle {\underline {\underline {C}}}}$. The upper-left 2x2 submatrix in ${\displaystyle {\underline {\underline {A}}}}$ represent a 45 degree rotation around the ${\displaystyle x_{3}}$ (or ${\displaystyle z}$) axis, plus and inversion through that same axis, i.e., ${\displaystyle x_{3}\rightarrow -x_{3}}$.