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I was trying to find out when a linear map is conformal (i.e. angle preserving) when I came across this.

These notes state that a linear map is angle preserving if and only if it is multiplication by a constant. But although I don't see where there is any mistake in the proofs it seems clear to me that rotations (which are not multiplication by a constant) also preserve angles.

Please could someone clarify to me when a linear map on $\mathbb R^n$ or $\mathbb C^n$ is conformal (=angle preserving)?

Anna
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2 Answers2

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$||L\mathbf v||=\lambda ||\mathbb v||$ does not mean that $L$ isn't a rotation. All it says is that every vector of unit length become length $\lambda$. It doesn't say anything about what $L$ does to the direction of any single vector.

Arthur
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At each $x $ in $\mathbf{R}^2$, $Df(x)$, i.e. the derivative of $f$ at $x$, is a matrix from $R^2$ to $C^2$. For $f$ to be conformal, at each $x$ this matrix has to be angle-preserving, thus, by the argument found in the link below, it has to be a multiple of an orthogonal matrix. But, notice that this will vary as $x$ changes.

ALL Orthogonality preserving linear maps from $\mathbb R^n$ to $\mathbb R^n$?

Community
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Behnam Esmayli
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