# 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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### Proof for Semidefinite symmetric matrix product

If A is symmetric positive semidefinite, show that: (a) For any matrix B,$BAB^T$is also positive semidefinite. (b) All the diagonal elements of A are nonnegative.
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### Minimization using Singular value

Let $A$ be a $p\times q$ matrix, with rank $q$. Show that the vector $x$ that minimizes $\|Ax\|_2$ under the constraint $\|x\|_2 = 1$ is the right singular vector of $A$ corresponding to the smallest ...
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### How to maximize matrix products

Find a unit vector v1 and a unit vector v2, such that the term: $$v^T \begin{bmatrix} 6 & -2 \\ -2 & 6 \end{bmatrix}v$$ is minimized and maximized, respectively. What are the minimum and ...
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### Multiplication of an orthogonal matrix by its first column

I am given a real orthogonal matrix Q (nxn), where the first column of Q is the vector x (nx1) where the 2-norm of x equals 1. I am asked to prove that QTx has first entry 1 and all the others zero: ...
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### Norms and equivalence classes question

Let $f\in C[0,1]$. Recall two of the norms we considered in class: $$\|f\|_\infty = \sup_{t\in[0,1]}|f(t)|, \quad \|f\|_1 = \int_0^1|f(t)|\ \mathsf dt.$$ Consider the space $C^1([0,1])$ of ...
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### lower bound on the minimum singular value of $\underline{\sigma} (A+B)$

what can we say about the lower bound on $\underline{\sigma}(A+B)$? Can we say the following? $\underline{\sigma}(A+B)>\underline{\sigma}(A)+\bar{\sigma}(B)$, where $\underline{\sigma}$ denotes ...
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### Can every polynomial be represented as a field norm?

Consider $f\in\mathbb Q[x_1,x_2,...,x_n],$ such that its degree is n and is irreducible.Can we find a normal extension $K/Q$ of dimension $n$ such that $$N_{K/Q}(\alpha)=f?$$
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### Relation between infinity norm and LU factorization

Let $A$ be a non-singular $n \times n$ matrix and suppose that Gaussian elimination with partial pivoting has been applied to produce $PA = LU$, where $P$ is a permutation, $L$ is a unit lower ...
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### Find the distance between two convex sets

Let's say that ||.|| is an Euclidian norm in $R^3$ and we have two sets in $R^3$ defined by inequalities: $Y = \{y| f(y)<a\}, Z = \{ z| g(z)<b \}$ Let's say that $f$ and $g$ are convex and ...
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### Variational characterization of nuclear norm

The nuclear norm $||\cdot||_{*}$ of a matrix is defined as the sum of its singular values. Working from the result at the bottom of this blog post, we have, for a matrix $\mathbf{X}$ and its ...
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### Do $\ell_{\infty}$ and $\ell_{2,1}$-norms give similar results? and why?

I am aware that $\ell_{2,1}$-norm is being used for inducing a structure in the sparse matrix. I was told by someone that $\ell_{\infty}$ and $\ell_{2,1}$-norms give similar results. But how and ...
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### What's a name of this quantity: $\max_i \big\lvert {\| x_i \|_2}^2 - 1 \big\rvert$?

What's a name of this quantity: $\max_i \big\lvert {\| x_i \|_2}^2 - 1 \big\rvert$? I defined this quantity to measure how the given set of real vectors is far from a set of normalized ones. Perhaps ...
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### Why is the *least-norm* solution useful or desirable?

For a problem at work, I had to write a program that would solve a system of linear equations. Often times the problem would lead to an underdetermined system (where there are less equations than ...
For the vectors $x$ and $y$, the Cauchy–Schwarz inequality reads $$|x\cdot y|\leq||x||\cdot||y||$$ Does this inequality only hold for 2-norm? Or for any norms? Thanks in advance.