# Why does an orthogonal matrix have the property $Q^T Q = Q Q^T = I$?

According to Wikipedia, an orthogonal matrix is a square matrix whose columns and rows are orthonormal vectors. It also says that this definition is equivalent to saying that an orthogonal matrix $$Q$$ is a matrix for which $$Q^T Q = Q Q^T = I$$, where $$I$$ is the identity matrix. Why are these definitions equivalent?

• If you write out matrix multiplication, it turns out that you’re computing a bunch of dot products of the columns of $Q$. – Michael Burr Dec 21 '18 at 14:07
• – Widawensen Dec 21 '18 at 14:46

## 2 Answers

The $$(i,j)$$ entry of $$Q^{T}Q$$ is the dot product of the $$i$$-th column of $$Q$$ with the $$j$$-th column of $$Q$$. Since $$Q$$ is orthogonal, these will be $$0$$ when $$i\neq j$$, and $$1$$ when $$i=j$$ (because the columns are unit vectors). Therefore $$Q^{T}Q$$ has ones on the diagonal and zeros everywhere else, so it's the identity matrix.

As pwerth noted, in the product $$Q^\top Q$$, the $$(i,j)$$-entry is simply the inner product of the $$i$$-th column of $$Q$$ with the $$j$$-th column. It is worth noting however that this implies that $$QQ^\top$$ is also the identity. This is because when a square matrix has a left-inverse, this left-inverse is immediately a right-inverse also.

The reason why this is interesting is because this makes a big use of the finite-dimensionality of the vector space we work over (in this case, $$\mathbb{R}^n$$, with $$n$$ the size of $$Q$$). For more details, see this question, where you'll also find a counterexample in the infinite dimensional case.