# Tagged Questions

Questions on the various algorithms used in linear algebra computations (matrix computations).

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### Is the condition number of unitary matrix always equal to 1?

I know that the 2-norm condition number $\kappa (\textbf U)={||\textbf U||_2}{||\textbf U^{-1}||_2}$ of a unitary matrix $\textbf U$ is always equal to 1. Is this true for all induced matrix norms, i....
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### why computing x+1 is stable? [on hold]

I understand it is not backward stable. If we go by definition of stability we get in the numerator Order(machine epsilon) but we have the denominator containing |x(1+delta)+1| where delta is Order(...
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### Finding a linear combination with constraints on coefficients

Let there be $n$ unit vectors $\{\boldsymbol{u}_i\}_{1\leq i\leq n }$ in an $m$ dimensional space. The vectors are not necessarily a basis of the space. Let $\boldsymbol{v}$ be a unit vector in the $m$...
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### Proving that Newtons Divided difference satisfies a particular formula

Assume $x\neq x_i$ for $0\leq i \leq n$ and show that the divided difference $f[x_0,\ldots,x_n,x]$ satisfies $$f[x_0,\ldots,x_n,x] = \sum_{i=0}^{n}\frac{f[x,x_i]}{\prod_{j=0,j\neq i}(x_i - x_j)}$$ ...
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### Linear regression of matrix elements to get the minimal polynomial to perform a matrix inversion?

So each matrix $\bf A$ fulfils an equation for it's minimal polynomial $P_m({\bf A})$: $$P_m({\bf A}) = 0 \Leftrightarrow \sum_{k=0}^{k_n}c_k{\bf A}^k = 0$$ We can by multiplying with $A^{-1}$ and ...
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### Compute a diagonalizable matrix close in matrix exponential

It is known that for any matrix $A$, one can perturb $A$ slightly so that the resulting $A(\epsilon)$ is diagonalizable. I am wondering whether for any matrix $A$, $\epsilon>0$, there is an ...
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### Derivatives of Matrices and Vectors

I am currently studying deep learning and a lot of the calculus involving differentiating products or sums of ill defined operations on matrices and vectors is very confusing. For instance, take ...
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### Singular value decomposition: does the choice of eigenvectors matter?

I'm trying to calculate the SVD-decomposition of a certain matrix, i.e. $A = U \Sigma V^T$. My solution doesn't yield $A$ again; I just can't get the signs correct. I'm wondering if this is just a ...
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### Machine Learning : Proof of equality

currently, I am writing on a paper that also makes use of machine learning techniques. My problem is as follows: I have binary classificator $h_w(\vec{x}^{(i)})$ that simply uses the sigmoid function, ...
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### Matrix similar and unitarily diagonalizable

Let $A,B \in R^{n \ x \ n}$ similar and unitarily diagonalizable. Prove that there $Q$ unitarily such that $Q^{H}AQ=B$
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### SVD as a solution to linear least squares

I'm a little confused about the various explanations for using Singular Value Decomposition (SVD) to solve the Linear Least Squares (LLS) problem. I understand that LLS attempts fit $Ax=b$ by ...
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### Task with using of A Linear Operator

Population of slithy toves living in severe and adverse conditions, subject to the following rules: a) On average, only half toves survive the first year of life, half of the remaining survives in ...
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### Create a matrix of four-dimensional space rotate counterclockwise by the angle π / 3 around the plane

Create a matrix of four-dimensional space rotate counterclockwise by the angle $\frac{π}{3}$ around the plane \begin{cases} x − y + t = 0,\\[2ex] y + z + t = 0 \end{cases} on the basis of unit ...
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### Undestanding SVM

I am the moment trying to understand how SVM works.. I understand the concept of finding a seperating hyperplane with the highest margin, but i do not understand how it works in mathmatically. Mor ...
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### Complex, symmetric linear equations

What is the fastest numerical method to accurately solve complex, symmetric linear equations? Preferably, with a link to a Fortran code. The dimension is about 100-200 variables. To be more explicit, ...
In the proof of Low-rank approximation by Trefethen & Bau, It is written: Theorem 5.8 : A is an $m \times n$ Matrix. For every $v$ with $0 \leqslant v \leqslant r$, define  A_{v}=\...
### Data structure for a symmetric $n\times n$ matrix
Suppose you are given a symmetric matrix $A\in\mathbb{R}^{n\times n}$ and consider the computation of the matrix vector product $A u \rightarrow v$ where $u\in\mathbb{R}^n$ is given and \$v\in\mathbb{R}...