# Tagged Questions

Norm is a function on a vector space $X$ which generalizes notion of length of vector in general vector spaces.

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### How to compute 2-norm between two matrices of different sizes?

I have a matrix $A$ of size $n\times8$ and matrix B of size $m\times8$. I need to compute $2$-norm to measure similarity. It can be measured as: $d(A,B)= \frac{||A-B||_{2}}{||A||_{2} ||B||_{2}}$
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### How to describe minimization of L1 norm error using linear programming?

Given a set of $n$ pair points $(x_1, y_1), ..., (x_n, y_n)$ in the plane, I need to find a line $ax + by = c$ that fits the points of the L1 norm error points as closely as possible. I need a linear ...
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### Operator norm under shrinkage

If I have a $n$-dim matrix $A=\{a_{ij}\}$, and I multiply each elements by a factor $w_{ij}$ in $[0,1]$, and get a new matrix $A_w=\{a_{ij}w_{ij}\}$. Do I have $$||A||\ge \lVert A_w\rVert$$ where the ...
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### If the entries of a positive semidefinite matrix shrink individually, will the operator norm always decrease?

Given a positive semidefinite matrix $P$, if we scale down its entries individually, will its operator norm always decrease? Put it another way: Suppose $P\in M_n(\mathbb R)$ is positive ...
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### Proof $\sigma_{\min}(A\Delta)\geq\sigma_{\min}(A)\sigma_{\min}(\Delta)$, $\sigma$ is a singular value

Let $A$ and $\Delta$ be square matrices. The definition of smallest singular value of a matrix $A$. (in title, $\sigma_{\min}$): The matrix norm is the 2-induced norm. The propertie: I don't ...
### Prove that $\bigg\| \begin{bmatrix} X \\ A\end{bmatrix} \bigg\|_2 \leq \sigma \iff X^* X + A^* A \preceq \sigma^2 I$
Prove that $$\bigg\| \begin{bmatrix} X \\ A\end{bmatrix} \bigg\|_2 \leq \sigma \iff X^* X + A^* A \preceq \sigma^2 I$$ Here, * denotes the conjugate transpose. This norm is the $2$-induced ...