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Given the trace of a Matrix $\boldsymbol A$, its trace is defined as $\mathrm{tr} (\boldsymbol A) = \sum \limits_i A_{ii}$

We can think of the trace as a scalar function $f(A_{11},\dots,A_{nn}) = A_{11} + \dots + A_{nn}$ from $\mathbb{R}^n$ to $\mathbb{R}$.

Is there an inverse function for $f$?

Based on the discussion in here, Can a scalar-valued multivariate function be invertible?, scalar functions can have an inverse function. But I do not know if this applies for the trace function - if it can be treated at all as a scalar function.

Simon
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    $f$ should rather be defined as $f(A)=A_{11}+...+A_{nn}$ where $A\in \mathbb R^{n\times n}$. And obviously, it's not invertible since it's not injective. – Surb Jul 25 '22 at 10:26

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Well, the simple answer is no unless your space is of dimension $1$. Otherwise you can always find different arguments/matrices having the same trace, thus you cannot revert the operation.

b00n heT
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  • Thank you! I would appreciate if you could help me out as for this question here as well: https://math.stackexchange.com/questions/4499738/derivative-of-scalar-function-w-r-to-another-scalar-function-with-the-argument – Simon Jul 25 '22 at 11:10