# Tagged Questions

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

110 views

63 views

### 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 ...
30 views

### Backward Euler method and the trapezoidal method are zero-stable

Show that backward Euler method and the trapezoidal method are zero- stable. I have no clue to prove the claim. Can anyone give me some hints? Thank you in advance. For someone who don't know ...
1k views

15 views

### Minimising two interdependent equations with least squares regression.

Originally, I had a set of points in three dimensional space that I was fitting using linear regression. So my model is $$Y = \alpha A+ \beta B$$ where $Y = \{y_i\}$ is the dependent variable, and ...
46 views

### Doubts on inverse power method

I found written that if matrix A is real and you use the Power method to find eigenvalues then "If the matrix and starting vector are real then the power method can never give a result with an ...
17 views

31 views

42 views

### 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 ...
45 views

### 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 $f^{-1}(E)$ ...
33 views

### 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 ...
48 views

### 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 ...
12 views

### 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 ...
33 views

### 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 ...
44 views

### 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
31 views

### 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 ...
38 views

### $LU$ 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 ...
26 views

### 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 ...
Assume that we have a real symmetric matrix $\mathbf{A}\in\mathbb{R}^{n\times n}$ obtained as following : $$\mathbf{A}=\mathbf{N}-\mathbf{P},$$ with $\mathbf{N}\in\mathbb{R}^{n\times n}$ and ...