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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?

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2 Answers 2

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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.

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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.

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