# Questions tagged [matrix-analysis]

For question about matrices and their algebraic properties. Together with [tag:linear-algebra] if necessary.

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### When $f\left(\|Ax\|\right)\leq \|f(A)x\|$ is true? [closed]

When is the following relation true? $$f\left(\|Ax\|\right)\leq \|f(A)x\|$$ where $A$ is any matrix and $\|\cdot\|$ is the spectral norm. Edit note : The function $𝑓$ is any real concave ...
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### Invertibility of infinite-dimensional matrix

I have a matrix $M \in \mathbb{R}^{n \times n}$ whose columns are linearly independent. Hence, $M$ is invertible. How to extend this conclusion to the case where $n$ is infinite? Specifically, ...
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### Do eigenvalues depend smoothly on the matrix elements of a diagonalizable matrix?

Suppose I have a matrix $M(t)$ whose matrix elements depend smoothly on a real parameter $t$. I also know that this matrix is diagonalizable for the $t$s I'm interested in. Can I say that its ...
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### Show that if $g(x) = f(Ax + b)$, then $\delta g(x) = A^T \delta f(Ax + b)$
Looking for a simplier expaination for the following: Show the following for sub-gradients: (a) If $g(x) = f(Ax + b)$, then $\delta g(x) = A^T \delta f(Ax + b)$. I've found the trivial ...