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Some definitions of orthogonal matrix (for instance, on Wikipedia) use the definition:

$$ Q^TQ=QQ^T=I $$

While other definitions (for instance, on Wolfram MathWorld) only include one of the products:

$$ QQ^T = I $$

I remember reading that $Q^TQ$ implies $QQ^T$ for orthogonal matrices, which would explain why some definitions don't list both of the products, but it is not immediately obvious to me why this is the case. From what I understand, an orthogonal matrix is a matrix whose columns are orthogonal unit vectors (i.e. $||z_i||_2^2 = 1$ and $z_i \cdot z_j = 0$ for $i \neq j$), which directly leads to the identity $Q^TQ = I$. However, I don't see why $QQ^T = I$ is necessarily true.

mai
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2 Answers2

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Remember that if $AB = I$ then $B=A^{-1}$ and so $BA = I$.

So if $QQ^T = I$ then $Q^T = Q^{-1}$ and so $Q^TQ=I$.

LucaMac
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  • First line is true when $A$ and $B$ are square. $Q$ is required to be square by definition, so second line follows from the first. – Alex Jones Sep 17 '18 at 22:06
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    Of course! That's what I meant! – LucaMac Sep 17 '18 at 22:08
  • Do both of the statements $AA^{-1} = I$ and $A^{-1}A = I$ always hold for an invertible matrix $A$? Is it possible to have one but not the other? – mai Sep 17 '18 at 23:51
  • Yes they do! That's because $\left(GL_n(\mathbb{K}), \cdot \right)$ is a group – LucaMac Sep 18 '18 at 12:01
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For an orthogonal matrix $Q$ we have

$$Q^{-1}=Q^T \iff QQ^T=Q^TQ=I$$

user
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