# Tagged Questions

Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically ...

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### Gradient and Hessian of function on matrix domain

Let $A \in R^{k \times p}$. Define $f(X) : R^{p \times k} \rightarrow R$ to be $f(X) = \log \det(XA + I_{p})$, where $I_{p}$ is a $p \times p$ identity matrix. I want to know what is the gradient and ...
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### Completing a semi-vector space to a vector space

If we have a semi-vector space $U$ (as defined here), what do we have to additionally demand from $U$ such that we can complete it to a vector space $\tilde{U}$ via $U\xrightarrow{\iota}\tilde{U}$ and ...
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### Maximum and minimum of determinant of matrices with entries from $\{0,1\}$ or $\{-1,0,1\}$

Maximal and Minimal value of $\bf{3^{rd}}$ order determinant whose elements are from the set $\bf{\{0,1\}}$. Maximal and Minimal value of $\bf{3^{rd}}$ order determinant whose elements are from the ...
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### $\dim C(AB)=\dim C(B)-\dim(\operatorname{Null}(A)\cap C(B))$

Let $A \in M_{n \times m}\left(F\right)$ and $B\in M_{m \times p}\left(F\right)$ for a field $F$. Prove: $\dim C(AB)=\dim C(B)-\dim(\operatorname{Null}(A)\cap C(B))$, where $C(X)$ denotes the column ...
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### Eigenvalue perturbation of singular matrix

Consider a Hermitian matrix $\mathbf{A_0} \in \mathbb{C}^{N \times N}$ with one singularity, i.e. its eigenvalues in increasing order are: 0 < \lambda_2 \leq \lambda_3 \leq \cdots \...
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### Why is this matrix symmetric?

There is an example in the Convex Optimization lecture notes, Boyd. He just said in the lecture that the matrix which is underlined in red color is symmetric! How can we claim that when there is no ...
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