Orthogonal Matrix

Definition

Let $A$ : Invertible

If $A^{-1}=A^T$ or equivalently, if $A{A}^T=A^{T}A=I$

Then we say $A$ is an orthogonal matrix

Alternative definition

• Invertible $A$ is orthogonal if $A^{-1}=A^T$, or equivalently if $A{A}^T=A^{T}A=I$
• Invertible matrix is said to be orthogonal, if its inverse is equal to its transpose.

Example

$$\left[\begin{array}{cc}\frac{\sqrt{2}}{2}& \frac{\sqrt{2}}{2}\\ -\frac{\sqrt{2}}{2}& \frac{\sqrt{2}}{2}\end{array}\right]\left[\begin{array}{cc}\frac{\sqrt{2}}{2}& -\frac{\sqrt{2}}{2}\\ \frac{\sqrt{2}}{2}& \frac{\sqrt{2}}{2}\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$