# Tagged Questions

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

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### stability function

I have an exercise which asks me to find polynomials $P$ and $Q$ with a degree $2$ that satisfy $$\exp(z)= \dfrac{P(z)}{Q(z)} + O(z^5)\ \text{for} \ z\to 0$$ My question is: Are they actually unique ...
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### Question about flipping terms in matrix multiplication in proving that $h(N_n(\mu , K))=\frac{1}{2}\log(2 \pi n)^n |K|$

So in my book, it is written: Let $X_1,X_2,...,X_n$ have a multivariate normal distribution with mean $\mu$ and covariance matrix $K$ and $\textbf{X}=(X_1,X_2,...,X_n)$ The above isn't really ...
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### numerical computation without explicitly calculating certain matrices

I have to numerically multiply: $A^{-1} B A$ where B is a diagonal square matrix, and A is symmetric. A is calculated from multiplying two non-square matrices, $A = XX^T$ I know B and X, and A and ...
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First of all, I´m very sorry for my bad english, especially writing. Ok, for differents problems i´m studing a Bachelor degree in Mathematics. These degree is online. Now, the problem with my school ...
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### Jordan Canonical form 2x2 matrix

Compute the Jordan Canonical form of A = $\begin{bmatrix}i & 1\\1 & -1\end{bmatrix}$. My (feeble) attempt: After I compute the characteristic polynomial, which gives me $x^2=0$, the ...
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### Nontrivial Matrix-estimate

I try to proof the following estimate: \begin{align} h' W^{-1} H W^{-1} h \geq c h' H h \qquad c>0, \qquad\qquad (1) \end{align} where $h\in\mathbb{R}^{K-1}$ and ...
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### Combine 2 sparse QR factorizations

I have sparse matrix $A_1$ which is size $m_1 \times n$ and another sparse matrix $A_2$ which is size $m_2 \times n$, where $m_1 < n$ and $m_2 \leq n$ and plan on stacking them to make a sparse ...
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### Set up for matrix solutions

I've haven't touched linear algebra in a while so I'm sorry if this seems simple but I did a google search and I am still confused. I have to find a solution to the following set of equations: ...
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### $l_1$ Matrix Norm Inequality

I am independently studying Numerical Analysis and came upon the following question: $l_1$ vector norm $||x_1||$ is defined as $||x_1||=\sum|x_i|$. How can we show that for the natural matrix ...
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### Richardson Iteration

Given the Richardson Iteration, $x_{n+1} = x_n + \alpha(b-Ax_n)$ (with $\alpha$ a scalar constant). To which polynomial $p(A)$ at step $n$ does this iteration correspond to? My first idea ...
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### Generalized minimal residuals: eigenvalues and sets of functions

Can someone help me on this exercise (2 parts)? Thanks! Suppose that $S \subseteq \mathbb{C}$ is a set whose convex hull contains $0$ in it's interior (so $S$ is contained in no half-plane ...
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### Numerical range of a matrix contains the convex hull of the eigenvalues.

I am stuck with the following question. Question: Let $A \in \mathbb{C}^{m \times m}$ be arbitrary. Let $W(A)$ be the numerical range i.e. the set of all Rayleigh quotients of $A$ corresponding to a ...
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### On eigenvalues, hermitian matrices and SVD

Are my ideas on the following "true or false"-statements correct? If $A$ is hermitian and $\lambda$ is an eigenvalue of $A$, then $|\lambda|$ is a singular value of $A$. My answer would be ...
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### Applications of Numerical methods

I'm in a course of Numerical Methods and part of an assignment is find an article about an application of numerical methods, explain this article and present a program (in matlab/octave) that ...
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### Backward stable algorithm

Assume we have fixed unitary matrices $Q_1, \dots, Q_k \in \mathbb{C}^{m,m}$ and a matrix $A \in \mathbb{C}^{m,n}$ which can be perturbed. How can we proof that the algorithm on computing the product ...
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### Avoid evaluation of a very large matrix in non-negative matrix factorization

This is somewhere in between a math and a programming question, so please send me back to SO if you think it's off-topic. I'm implementing non-negative sparse coding, a regularized variant of ...
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### Inverse of Sum of Matrix Inverses

Given $N$ positive-definite matrices $\Lambda_i$, I need to efficiently compute $\Gamma_N$, where $$\Gamma_n = \left(\sum_{i=1}^n \Lambda_i^{-1}\right)^{-1}.$$ Applying the Woodbury matrix identity ...
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### Range and kernel of a linear transformation are ALWAYS disjoint

Is it true that the Range and kernel of a linear transformation are ALWAYS disjoint. I think they are not but I remember in my notes that the ker L= Im (L') this was under projections. So I am unsure ...
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### Conditional number: exercise

Let's say we've got a $202 \times 202$ matrix $A$ for which $||A||_2=100$ and $||A||_F=101$ (the Frobenius norm). How can we find the sharpest bound (lower) on the 2-norm condition number of $A$? ...
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### How to condense a matrix to a vector

I'm not an experienced person in mathematics and this might either sound like a trivial question or a stupid one. However, this problem arose to me when I was writing a program. Following is my ...
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### Proving an identity

We define $\|x\|_A^2:= x^TAx$ and $(x,y)_M := y^TMx$ for a symmetric positive definite matrix $A$ and an invertible matrix $M$. I want to show the following identity for the errors of Richardson's ...