Equivalence of Definitions of Orthogonal Matrix

Theorem

The following definitions of the concept of Orthogonal Matrix are equivalent:

Definition $1$

Then $\mathbf Q$ is orthogonal if and only if:

$\mathbf Q^{-1} = \mathbf Q^\intercal$

where:

$\mathbf Q^{-1}$ is the inverse of $\mathbf Q$

$\mathbf Q^\intercal$ is the transpose of $\mathbf Q$

Definition $2$

Then $\mathbf Q$ is orthogonal if and only if:

$\mathbf Q^\intercal \mathbf Q = \mathbf I$

where:

$\mathbf Q^\intercal$ is the transpose of $\mathbf Q$

$\mathbf I$ is the identity matrix of the same order as $\mathbf Q$.

Definition $3$

Then $\mathbf Q$ is orthogonal if and only if:

$\mathbf Q = \paren {\mathbf Q^\intercal}^{-1}$

where:

$\mathbf Q^\intercal$ is the transpose of $\mathbf Q$

$\paren {\mathbf Q^\intercal}^{-1}$ is the inverse of $\mathbf Q^\intercal$.

Proof

Definition $(1)$ is equivalent to Definition $(2)$

\(\ds \mathbf Q^{-1}\)

\(=\)

\(\ds \mathbf Q^\intercal\)

Definition 1 of Orthogonal Matrix

\(\ds \leadstoandfrom \ \ \)

\(\ds \mathbf Q^{-1} \mathbf Q\)

\(=\)

\(\ds \mathbf Q^\intercal \mathbf Q\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \mathbf I\)

\(=\)

\(\ds \mathbf Q^\intercal \mathbf Q\)

Definition 2 of Orthogonal Matrix

$\Box$

Definition $(1)$ is equivalent to Definition $(3)$

\(\ds \mathbf Q^{-1}\)

\(=\)

\(\ds \mathbf Q^\intercal\)

Definition 1 of Orthogonal Matrix

\(\ds \leadstoandfrom \ \ \)

\(\ds \paren {\mathbf Q^{-1} }^{-1}\)

\(=\)

\(\ds \paren {\mathbf Q^\intercal}^{-1}\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \mathbf Q\)

\(=\)

\(\ds \paren {\mathbf Q^\intercal}^{-1}\)

Definition 3 of Orthogonal Matrix

$\blacksquare$