# Tagged Questions

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

4answers
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### How to find 2x2 matrix with non zero elements and repeated eigenvalues?

I need to find a 2x2 matrix with non zero elements that has eigenvalue = 1 repeated (double). How can i do that? Thanks!
1answer
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### Computing the best-fit plane normal from n points

I've been working steadily through "3D Math Primer for Graphics and Game Development" and am stuck on how the authors derived their equation for the best-fit plane normal given n points. Please note, ...
0answers
177 views

### Difference between Householder Reflections and Gram-Schmidt?

In numerical QR decomposition, when we calculate the orthonormal factor Q of a matrix, what is the difference in results if we use Householder Reflections to normalize the matrix or use Gram-Schmidt ...
2answers
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### Can you compute rank r factorization of a n*n matrix in time O(n^2 r)?

I am wondering if you can compute the SVD/eigenvectors of a rank r matrix of size n*n in time O(n^2 r)? My understanding is that standard eigenvector computations involve bringing matrix into ...
2answers
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### How to understand or show this?

We have $$F=ABh^{p_1}+\theta (h^{p_2})$$ $$G=Ah^{p_1}+\theta (h^{p_2})$$ We $A$,$B$ are real numbers, $h$ positive, $|h|\leq 1$ , $p_1<p_2$ natural numbers and $\theta(h)$ means that it is of ...
1answer
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### Rate of Convergence of Generalized Iterative Method

Consider the generalized iterative method for finding polynomial roots: $z_{k+1}=z_k +d\frac{(1/p)^{(d-1)}(z_k)}{(1/p)^{(d)}(z_k)}$ where d is a positive integer. Note that Newton's Method is a ...
3answers
134 views

### properties of positive definite matrix

If $A$ is a symmetric positive definite matrix can we conclude $A^{n}$ is positive define too? Why? For example for $n=2$: $x^{T}(AA)x=x^{T}(AA^{T})x=(x^{T}A)^2>0$; for $n>2$?
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### function of matrix and eigen values

I want to calculate exp(A), A is matrix, with numann series. is this series depend of matrix's eigen values? for example if it's eigen values are large, is numann series useful for this function?
1answer
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### Differences between methods for solving linear equation system

I have a huge linear equation system in this form: F=K.Δ as usual form of problems in the finite element method, where the F vector and K are known and Δ vector is unknown. There are several ...
1answer
217 views

### Need matlab help to construct a numerical example for solving system of linear equation for random matrices

I am reading this paper(page 183). In this paper the iterative methods for computing some solution of the general restricted linear equations \begin{eqnarray} Ax = b, ~~~~ x\in R(A^{k})~~~~ b\in R(A^{...
1answer
93 views

### Most efficient way to find distinct complementary subspaces over a finite field

Let $V$ be a $n$-dimensional vector space over $\mathbb{F}_p$ and let $W$ be a $k$-dimensional subspace. What's the most efficient way to algorithmically write down a basis for each distinct subspace ...
1answer
114 views

### Understanding higher order SVD

Can someone explain the singular value decomposition of a tensor (maybe a 3 dimensional matrix) with an example? It is intuitively difficult to the get the meaning from just the formulas. On a ...
2answers
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### A linear programming to obtain “canonical basis of convex cone”

In my research a I need to solve the linear equation (getting its null space) under some constraints. The matrix is given below: The constraints shall be (x1...x[16]>0, x[17]...x[20] arbitary...) ...
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### Proof $\langle Ax,y\rangle = \langle x,A^*y\rangle$ when $A$ Hermitian

I was trying to understand a proof of why a Hermitian $A$ matrix has its eigenvectors orthogonal. As part of the proof they state $$\langle Ax,y\rangle = \langle x,A^*y\rangle$$ From which property ...
0answers
329 views

### Solving linear system of equations using Successive Over-Relaxation

I was solving a system of linear equations with SOR. I used different values of relaxation factor (w) for the different runs. I found that for all w > w' (1 < w' < 2), the error is the result ...
0answers
247 views

### Orthogonal Procrustes Problem

The classical orthogonal Procrustes problem concerns finding the matrix $\Omega$ which minimizes $||A\Omega-B||_{F}$ subject to $\Omega'\Omega=I$, with A and B known matrices. Let A be the identity. I ...
1answer
260 views

### Solving Poisson Equation Finite-difference using maple

How do I solving Poisson Equation Finite-difference using maple consider Poisson equation $$\frac{\partial^2u}{\partial x^2} (x,y)+ \frac{\partial^2u}{\partial y^2} (x,y) = x*e^y$$ $0<x<2$ ,...
1answer
199 views

### QR factorization of a special structured matrix

A friend asked me the following interesting question: Let $$A = \begin{bmatrix} R \\ \xi{\rm I} \end{bmatrix},$$ where $R \in \mathbb{R}^{n \times n}$ is an upper triangular and ${\rm I}$...
1answer
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### Are Similar Matrices and Unitary Property related?

Recall that 2 matrices $A, B\in R^{n,n}$ are similar if there exists a matrix $P$ such that $A=P^{-1}BP$. In this case is $P$ always orthogonal?
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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 ...
1answer
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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 ...
2answers
323 views

### Advice in Bachelor Degree

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 ...
1answer
550 views

### 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 ...
0answers
26 views