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Consider the function $V:\mathbb{R}\to\mathbb{R}$ given by

$$ V(t)=\|I - e^{At}\|^2 $$

where $I$ is the identity matrix and $A$ is a square matrix. The norm is the Euclidean norm on $M_n(\mathbb{R})$:

$$\|X\|=\sqrt{\lambda_{max}(X'X)};\ X\in M_n(\mathbb{R})$$

that is induced by the matrix norm $\|x\|^2=x'x$ on $\mathbb{R}^n$.

I want to calculate the derivative $\frac{dV(t)}{dt}$. Is this possible in any way?

Note: Maybe the use of the Frobenius norm would facilitate things a bit but I wouldn't prefer it as it is not an induced norm (by some matrix norm).

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