# Linear algebra/Orthogonal matrix

Alternative notations
$Q^{-1}=Q^{\mathrm {T} }$ $Q^{\mathrm {T} }Q^{-1}=Q^{-1}Q^{\mathrm {T} }=I$ ${\underline {\underline {Q}}}^{-1}={\underline {\underline {Q}}}^{\mathrm {T} }$ ${\underline {\underline {Q}}}^{\mathrm {T} }\cdot {\underline {\underline {Q}}}^{-1}={\underline {\underline {Q}}}^{-1}\cdot {\underline {\underline {Q}}}^{\mathrm {T} }={\underline {\underline {I}}}$ $Q_{jk}^{-1}=Q_{kj}\equiv Q_{jk}^{\mathrm {T} }$ $\sum _{k}Q_{ki}Q_{kj}^{-1}=\sum _{k}Q_{ik}^{-1}Q_{jk}=\delta _{ij}$ A real-valued and square matrix $Q$ is orthogonal (or orthonormal) if $Q^{\mathrm {T} }=Q^{-1},$ where $Q^{\mathrm {T} }$ is the transpose and $Q^{-1}$ is its inverse. Equivalently, $Q$ is orthogonal when $Q^{\mathrm {T} }Q=QQ^{\mathrm {T} }=I,$ where $I$ is the identity matrix. It has a number of well-known properties:

### 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, ${\underline {\underline {C}}}={\underline {\underline {A}}}\cdot {\underline {\underline {B}}}:$ $\overbrace {\begin{bmatrix}{\frac {1}{\sqrt {2}}}&{\frac {1}{\sqrt {2}}}&0\\-{\frac {1}{\sqrt {2}}}&{\frac {1}{\sqrt {2}}}&0\\0&0&-1\end{bmatrix}} ^{\underline {\underline {A}}}\cdot \overbrace {\begin{bmatrix}1&0&0\\0&{\frac {1}{\sqrt {2}}}&{\frac {1}{\sqrt {2}}}\\0&-{\frac {1}{\sqrt {2}}}&{\frac {1}{\sqrt {2}}}\end{bmatrix}} ^{\underline {\underline {B}}}=\overbrace {\begin{bmatrix}{\frac {1}{\sqrt {2}}}&{\frac {1}{2}}&{\frac {1}{2}}\\-{\frac {1}{\sqrt {2}}}&{\frac {1}{2}}&{\frac {1}{2}}\\0&{\frac {1}{\sqrt {2}}}&-{\frac {1}{\sqrt {2}}}\\\end{bmatrix}} ^{\underline {\underline {C}}}$ Both ${\underline {\underline {A}}}$ and ${\underline {\underline {B}}}$ posses a symmetry that could lead one to postulate a non-existent symmetry among off-diagonal elements. But no such symmetry exists for ${\underline {\underline {C}}}$ . The upper-left 2x2 submatrix in ${\underline {\underline {A}}}$ represent a 45 degree rotation around the $x_{3}$ (or $z$ ) axis, plus and inversion through that same axis, i.e., $x_{3}\rightarrow -x_{3}$ .