How can we show the difference between an orthogonal matrix and an invertible matrix?

Question

This is probably a matter of definition, but I would like to get more insight in the differences between orthogonal matrix, invertible matrix and orthonormality

Given an orthogonal matrix $U$:

$$ U \in \mathbb{R}^{n\times n} $$ is orthogonal, then $$ U^{-1} = U^\top $$

How can we show the difference between an orthogonal (which is automatically invertible) matrix and an invertible matrix (which is not automatically orthogonal)? I think I have trouble making the difference given the fact that the rows (respectively rows) of an invertible matrix have already to be orthogonal

Moreover, respect to orthonormality, Can we speak about orthonormal matrix or can we only speak about orthonormal vector basis which form a orthogonal matrix?

Answer 1

An orthogonal matrix is invertible by definition, because it must satisfy $A^TA=I$.

An invertible matrix need not be orthogonal: consider $$ A=\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} $$ which is not orthogonal, but is invertible, with $$ A^{-1}=\begin{bmatrix} 1 & -1 \\ 0 & 1 \end{bmatrix}\ne A^T $$

In an orthogonal matrix the columns are pairwise orthogonal and each is a norm $1$ vector, so they form an orthonormal basis.

Answer 2

Every column in the orthogonal matrix is orthogonize to other columns, but this fact is not exactly hold for any invertible matrix.