# 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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### LU decomposition with pivot

I'm trying to LU decompose, with pivoting, the following matrix ($A=(a_{ij})$): A = [2 1 2; 1 0 3; 4 -3 -1]; % matlab I cannot make out from my literature (...
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### Inverted pendelum Matrix numerical derivative

Here I've written a dynamic function as : ...
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### how to calculate spectrum of a large collumn stochastic matrix

Okay, I have a collumn stochastic matrix of order $280\times 280$, the entries are given in an url in some webpage in row format. I need to find all the eigen values and eigen vector corresponding to ...
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### what's the difference between Eisenstat trick and an implicit preconditioner?

Assume ${A}$ is Hermitian positive definite and $\hat A$=$D^{-1/2}$$A$$D^{-1/2}$ is to obtain a symmetric variant. and $M$=($L_{A}$+$D$)$D^{-1}$($D$+$U_{A}$) where $D$ is a suitable diagonal matrix ...
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### Square-root of a matrix which arise from truncating a matrix which has a square-root

I have this covariance matrix $A$ which has a symmetric Toeplitz structure. A = \left[ \begin{array}{cccccccc} c_0 & c_1 & c_2 & \cdots & c_{n-1} & c_{n} \\ c_1 &...
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### Permutation of the posson equation

I have the poisson matrix A, I want to reduce it to the following block matrix: $$A=\begin{bmatrix} A_{11} & A{12} \\ A_{21} & A_{22} \end{bmatrix}$$, where $A_{11}$ and $A_{22}$ are ...
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### Inverse power method - how to decide which $\alpha$ to be used?

Inverse power method - how to decide which $\alpha$ to be used ? I've learnt who inverse power method run, but I don't know how to choose the $\alpha$ $y=(A-\alpha I)^{-1}u_k$ the $\alpha$ here
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### Prove $\left \| f-L_{k}^{s}f \right \|_{2} = min_{q \epsilon V_{k} }\left \| f-q \right \|_{2}$

Let q be arbitrary and consider the quadratic function of t defined by: $\phi (t)=\left \| f-L_{n}^{s}f+tq \right \|_{2}^{2}$ Note: $L_{k}^{s}f = \sum_{i=1}^{k}(f,p^{i})p^{i}$ for $i = 1,...,k$ ...
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### Generalized conjugate residuals method applied to a block matrix

I have the diagonal block matrix A with $2 \times 2$ k-blocks given by : $D_k=\begin{bmatrix} 1 & k\\ 0 & 1 \end{bmatrix}$. I have to show that the generalized ...
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### Conjugate gradient method and rietz values

I'm working on the conjugate gradient method. I have the matrix A, defined as A= diag(v) where $v=[ones(1,10), 11:1000]$. I have to solve the system $Ax=b$ ,b=ones(1000,1) with the conjugate ...
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### Find the inverse of $A+uB+vC+uvD+u^2E+v^2F$ where $A,B,C,D,E,F$ are symmetric.

Given scalars $u,v$ s.t. $0<u,v<1$, we seek the properties of the matrix defined by $$P=A+uB+vC+uvD+u^2E+v^2F$$ A is symmetric and positive definite. $B,C,D,E,F$ are symmetric, but might not ...
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### How to find How to find ΔU if LU factorization of tridiagonal Matrix A is used to sovle system Ax=b?

How to find ΔU if LU factorization of tridiagonal Matrix A is used to solve system Ax=b? By using forward and back substitution to show that x̂ satisfies (L̂+ΔL)...
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### Improving my QZ-Algorithm (Include Shifts)

I Need to to solve an generalized Eigenvalue Problem and compare two Methods (QR and QZ) concerning their convergence rate and execution time. I started with the Basic QR-Algorithm, implemented in ...
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### Rapid calculation of eigenvectors for a submatrix when I know the eigenvectors of the full matrix

I have an $N \times N$ matrix $Z$ that is complex-valued and nonsymmetric in general. I can solve for the eigenvalues and eigenvectors of this matrix numerically. Call the diagonal eigenvalue matrix ...
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### Extracting I-Vectors from a GMM-supervector

The probability model for I-vector in GMM-UBM system is : M = M(ubm) + Ty where, M is GMM supervector (means of all gaussian components concatenated in single vector) M(ubm) is UBM supervector I ...
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### Response Matrix with finit actuator

I have a system of penalties ($P$) and actuators($A$). Whereby: d$P_i/$d$A_j$ = close to constant $\quad\forall i,j$ In order to minimize $P$, I create a response Matrix ($M$). With its pseudo-...
I'm looking for some references on the complexity for the following kind of problem: Given two Basis $(a_1, ... ,a_n)$ and $(b_1, ..., b_n)$ of the $K(x)$-vector space $V$ I want to compute the ...