Tagged Questions

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

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Is $\| \sum_{i \in [k]} \otimes^3 v_i - T \|_F^2 + \theta \| \sum_{i \in [k]} \otimes^3 v_i \|_F^2$ convex?

I am trying to find the minima of the following equation with respect to $v_i$, $i \in [k]$, to solve an optimization problem but I can't manage to make (stochastic or not stochastic, neither of them) ...
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Prove Norm Theorems

I have the following as given: Let $A \in C^{m\times m}$. Then: 1) $$\lVert A\rVert_1 =\sup_{v\in C^m \setminus\{0\} }{\lVert A_v\rVert_1 \over \lVert v\rVert_1} = \max_{j} \sum_i |a_{ij}|$$ How can ...
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show that $\lambda_{max}(B^{-1}A) \leq 1$ (eigenvalues, matrices, preconditioning)

I'm trying to get more familiar with eigenvalues and matrices but struggle to see the following: $\lambda_{max}(B^{-1}A) \leq 1$ $A \in \mathbb{R}^{n \times n}$ is symmetric, positive definite ...
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upper bound of a function $n^{1/\log(n)}$

I have the following expression $n^{1/\log(n)}, \quad where \quad n \in [1, 10,000]$. When I solve this numericall, I get the resultant value 2.718282 for all $n \in [2, 10,000]$. On this basis, I can ...
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An exercise with linear maps

I'm solving the following exercise: $\newcommand{\RR}{\mathbb{R}}$ Let $f: \RR^3 \rightarrow \RR^4: (a,b,c) \mapsto (a+2b+c, b+c, a + 2b + 3c, a + b + 2c)$ Find the basis of $f^{-1}(E)$ ...
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Choosing value of ω for SOR

I am learning about successive overrelaxation, and I'm wondering if there is an intuitive reason as to why ω must be between 0 and 2. I know that the method will not converge is ω is not on this ...
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Dynamically equivalent of numerical solution of $y' = f(y)$.

Consider an autonomous system $y' = f(y)$ and a fixed step size $h$. a) Show that the trapezoidal method applied $N$ times is equivalent to applying first half a step of forward Euler, (i.e forward ...
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matrix optimization problem techniques

I'm looking for some resources on learning techniques commonly used in matrix optimization. For example, minimization of the Frobenius/nuclear/weighted norm of a function of a matrix subject to ...
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Unitary diagonalization of matrices

Can someone tell me whether every square matrix $A\in \mathbb{R}$ unitarily diagonizable? If yes what is a necessary and sufficient condition for a square matrix $A\in \mathbb{R}^{n\times n}$ to be ...
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Beginner Linear Algebra 1 equation with 3 variables

I do not understand what to put into the remaining values. I tried to solve for the y and z like I did for the x, but the system is telling me that is incorrect. Some help would be appreciated
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Is there any Eigen value decomposition, which can be warm-started?

I have a Matrix, $A$ which is positive semidefinite. No consider, $B=A+\Delta$. I have Eigen decomposition of $A$ and $\Delta_{ij}<= \epsilon_1$, $\Vert \Delta \Vert_F <= \epsilon_2$. Is there ...
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$LU$ Factorization

Suppose the $A\in\mathbb{R}^{n\times n}$ is nonsingular and that $A = LU$ is its $LU$ factorization. Give an algorithm that can compute, $e_i^TA^{-1}e_j$,i.e., the $(i,j)$ element of $A^{-1}$ in ...
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Is it true that solving a triangular system using forward or backward substitution numerically stable?

The system is $TX = B$, where $T$ is a triangular matrix, $X$ is a unknown matrix, and $B$ is the RHS matrix. I know the system $Tx = b$ is backward stable where $b$ is a RHS vector. Detail check ...
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Computing a new inverse given information from an earlier inverse:

Suppose I have linearly independent set of vectors $v_1 ... v_k \in \mathbb{R}^{k}$. I can let $B = [v_1 ... v_k]$ and by applying Gaussian elimination on the matrix $$[ B \ I_k]$$ I end up ...
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Comparing matrix norm with the norm of the inverse matrix

I need help understanding and solving this problem. Prove or give a counterexample: If $A$ is a nonsingular matrix, then $\|A^{-1}\| = \|A\|^{-1}$ Is this just asking me to get the magnitude of ...
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Conditioning in regards to matrix vector product

This program involves the matrix-vector computational primitive $y \leftarrow Ax$ where $x,y\in\mathbb{R}^n$ and $A\in \mathbb{R}^{n\times n}$. A is taken to be dense and banded in the two parts of ...
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Sparse Matrix or Dense Matrix

My task is to implement the inner product and vector triad forms for a dense $A$ in single and double precision. I have successfully implemented the inner product and vector triad form although, I am ...
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Eigenvalue perturbation theory for $(A^TA)(B^TB)^{-1} + (B^TB)(A^TA)^{-1}$

Let $A, B$ be $n \times n$ matrices with full rank. I'm interested in getting a bound on how the smallest eigenvalue of $S = (A^TA)(B^TB)^{-1} + (B^TB)(A^TA)^{-1}$ changes when I perturb $A$ and $B$. ...
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Order of $LU$ factorisation

Can someone tell me how to calculate the order of a) $LU$ decomposition as well as b)the gaussian elimination of a square matrix $A$? I am at a loss ... Given:: $A$ is a $n\times n$ matrix and ...
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Finding linearly independent columns of a matrix when $m < n$

I need to a maximal set of linearly independent columns of a matrix $A$. I've googled a lot and seen various solutions, but none of them seem to work for me. What I've seen so far is 1.- Using Cauchy-...
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Efficient Row Sum of Factorized Matrix

I am currently computing the row sums of a reduced rank factored matrix by reconstructing a row subset of the original (approximated) matrix. The matrix was factored using SVD: A -> U, S, V -> U, SxV ...
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Backward Stability Lemma

Lemma-Let $x\in \mathbb{R}^n$ and $y\in \mathbb{R}^n$ with components, $\xi_i$ and $\eta_i$, $1\leq i\leq n$, respectively, that are floating point numbers. Computing the inner product $x^Ty$ on a ...
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In common tongue, what is the differences between sparse and dense matrices?

What are the differences with sparse and dense matrices in practice, so as to offer some insight to new learners on a more intuitive level. Obviously everyone knows about the dictionary definition of ...