Assume $A$ is a matrix, and $I$ is the identity matrix. I want to measure how close $A$ is to $I$ such that the same measure doesn't hold for any other matrix.
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The determinant shows how the area of a polygon scales if the vertices are transformed by $A$. However, rotation matrices have the same determinant as identity. But they are not identity matrices.
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The matrix norm shows that largest scaling of a vector under $A$: $||A||=\max_{x\neq 0} ||Ax||/||x||$. Again, rotation matrices will give the same measure as identity.
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The Frobenius norm is the squared sum of elements of $A$. I can make up any number of matrices whose Frobenius norm is the same as that if $I$
Is there a similarity measure $M: A \rightarrow \mathbb{R}$ between a marix $A$ and identity $I$ such that $M(I) = M(A) \iff I=A$?
Best Answer
The function $$M(A)=\|I-A\|$$ satisfies your conditions. It's a somewhat natural choice, and can be used, for example, in a proof of the Jordan-Schur theorem.