0
votes
0answers
5 views

Stieltjes transformation of e.d.f. sample eigenvalues

If the eigen-decomposition of sample covariance matrix is $S=PDP'$ where $D$ is a diagonal matrix with eigen value of $S$ and $P$ are eigenvectors. If we define the empirical distribution function of ...
0
votes
1answer
31 views

Finding Eigenvalues with Gershgorin-Discs

$A=\begin{pmatrix}-5 & 0 & 0 \\ 2 & 2 & 1 \\ 3 & -5 & 4\end{pmatrix}$ Find the Gerschgorin-discs, where the eigenvalues of $A$ lie. According to the formula if we ...
0
votes
0answers
21 views

Diagonal Pivoting Algorithm

Commonly in LU factorization, partial pivoting is used. I know there is another pivoting which is diagonal pivoting. However, on the internet very few resources discussing diagonal pivoting (Only ...
1
vote
2answers
50 views

Exponential of a 3x3 lower bidiagonal matrix

I have a 3x3 matrix with non-zero entries ONLY along the main diagonal and the diagonal above. There are exactly two non zero diagonals in the matrix like this \begin{pmatrix} a & 0 & 0 \\ d ...
1
vote
2answers
32 views

Square Idempotent matrix: efficient algorithms for finding eigenvectors

Given a square idempotent $N \times N$ matrix $A$ with large $N$, and a priori knowledge of the rank $K$, what is the most efficient way to compute the $K$ eigenvectors corresponding to the $K$ ...
0
votes
0answers
45 views

Condition of a matrix proof

I have to show the following inequality: For given two invertible matrices $A,B \in \mathbb{R}^{n\times n}$ show that $k(AB)\leq k(A)k(B)$, where $k(A)=\left \| A \right \|\left \| A^{-1} \right \|$ ...
3
votes
4answers
78 views

Prove that for every vector $V$, $||V||_{\infty} \leq ||V||_2 \leq || V||_1$

$\newcommand{\inf}{||V||_\infty}$ $\newcommand{\two}{||V||_2}$ $\newcommand{\one}{||V||_1}$ Prove that for every vector $V$, $\inf \leq \two \leq \one$ I have tried to look online for a solution to ...
0
votes
0answers
5 views

Singularity check for Homographies

I know that the standard singularity check for a matrix represented in some finite-precision format (IEEE-754 or whatnot) is "the matrix is singular if the reciprocal of the condition number of the ...
2
votes
1answer
61 views

Quick way of finding the eigenvalues and eigenvectors of the matrix $A=\operatorname{tridiag}_n(-1,\alpha,-1)$

Matrix $A=\operatorname{tridiag}_n(-1,\alpha,-1)$ has the eigenvalue: $\lambda_i=\alpha-2\cos(i\theta),$ $i=1,\dots,n$ and the corresponding eigenvectors are: ...
1
vote
1answer
30 views

condition number after scaling matrix

Maybe a well-known question. Let $\Sigma$ represent a real symmetric positive definite matrix, i.e. a covariance matrix. Which diagonal matrix $D$ with positive diagonal minimizes the condition ...
4
votes
4answers
133 views

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!
1
vote
0answers
104 views

Minimum L1 norm may not obtain the sparsest solution?

Here is a paper called For Most Large Underdetermined Systems of Equations, the Minimal L1-norm Near-Solution Approximates the Sparsest Near-Solution. However, I did not quite get its definition of ...
0
votes
0answers
57 views

A=UL Doolittle method

A - symmetrical positively defined, matrx, 3-diagonal Make a modified Doolittle method A=UL U - upper triangular matrix L - lower triangular matrix with ones on the main diagonal I have to work on ...
0
votes
1answer
53 views

Solving Linear Systems by hand

My professor said for our final we would have to solve linear systems by hand on our final. Some of our questions for interpolation and finding splines involve large 6x6 or 12x12 matrices. What is the ...
1
vote
1answer
146 views

cube root of positive definite matrix

Suppose that $A$ is a real symmetric positive definite $20\times 20$ matrix with condition number $\kappa\le 1000$. I want to solve the system of linear equations $$A^{1/3}x=b$$ with $10$-digit ...
0
votes
0answers
76 views

Tridiagonal system problem with Gaussian elimination and partial pivoting

What happens to the tridiagonal system (shown below) if Gaussian elimination with partial pivoting is used to solve it? In general, what happens to a banded system? $$ \pmatrix{d_1 & c_1 & ...
0
votes
0answers
44 views

Calculating the Jacobian for sampled data

I am reading about the Jacobian matrix which I interpret as a generalized gradient. I would like to take my investigations a bit further, so I have constructed some sample data which I would like to ...
2
votes
0answers
61 views

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 ...
1
vote
1answer
35 views

an estimate for condition number: $\kappa(C^{-1}A)\leq \kappa(C^{-1}B)\kappa(B^{-1}A)$

I'm currently reading through "Domain Decomposition Methods" by Tosseli and Widlund and in the appendix I found the following Theorem: Let A, B, C be symmetric positive definite matrices. Let ...
1
vote
1answer
212 views

Natural Cubic Spline 3 points

I am trying to do a natural cubic spline but I'm having trouble. f(-.0247500)=-.5, f(.3349375)=-.25, f(1.101000)=0 I tried doing the matrix, Ax=b where, h0=h1=.25 an a0=-.0247500, a1=.3349375, ...
0
votes
0answers
173 views

Exact inversion of matrix complexity (by Gaussian elimination)

I would like to check if what I have done is correct. Please, any input is appreciated. Problem statement: Consider a non-singular matrix $A_{nxn}$. Construct an algorithm using Gaussian elimination ...
0
votes
0answers
33 views

How to generate random matrices when it's singular values are given?

Consider matrix S as nxn diagonal matrix with singular values populated across the diagonals in non-increasing order. I want to know how to create random matrix A whose singular values with be the ...
3
votes
1answer
54 views

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 ...
1
vote
0answers
310 views

Finding error and proving Romberg Integration Method

Let $f$ be a function, which its integral has to be approximated by using romberg method.The $n\cdot m$th cell in romberg matrix (a lower triangular matrix) is given by ...
0
votes
1answer
45 views

Condition Number of Polynomial (Condition Number = 0)

I'm calculating the condition number of a polynomial equation $$ y = (x-2)^{9} $$ for this equation, the Jacobian is equal to ...
1
vote
2answers
52 views

LU decomposition by hand

Can someone show me a step by step solution to calculate the $LU$ decompisition of the following matrix: $A = \begin{bmatrix} 5 & 5 & 10 \\ 2 & 8 & 6 \\ 3 & 6 ...
0
votes
1answer
111 views

Computational efficiency using Gaussian elimination

Assume it took 2 seconds to solve an equation Ax=b for x (where A is a 3×3 matrix and b is a 3×1 matrix) using Gaussian elimination, how much longer would it take to: a) use Gaussian elimination to ...
1
vote
1answer
116 views

How do deal with a giant sparse matrices?

Someone point me in the right direction. I'm looking to do some heavy-duty manipulation of some really large and often very sparse matrices. Naturally, this problem overlaps programming heavily (I ...
1
vote
1answer
125 views

Power iteration provably works if the matrix has a unique eigenvalue $\lambda$ and $\lambda>0$

Let $A$ be a $n\times n$ real matrix and $v_0 \in \mathbb R^n$ s.t. $||v_0|| = 1$. Define a sequence $(v_k)_k$ of $n$-dimensional real vectors by $v_k = A^kv_0 / || A^kv_0 ||$. Assume that $A$ has a ...
1
vote
0answers
135 views

Algebraic ellipsoidal least squares fit

I'm looking to perform a least squares fit in 3D to a quadratic surface of the form: \begin{equation} Ax^2 + Bxy + Cxz + Dy^2 + Eyz + Fz^2 + ax + by + cz + d = 0 \end{equation} by minimizing ...
1
vote
0answers
104 views

Convergence of the Jacobi iteration method

I think I am not quite understanding the Jacobi Method or some related concept for indirectly solving linear systems of equations of the form $Ax=b$. We need the norm $||I-Q^{-1}A||_\infty < 1$ and ...
0
votes
1answer
147 views

approximating diagonal of inverse sum of low rank and diagonal matrices

I was wondering if there is any theorem or algorithm to approximate the diagonal elements of the inverse of sum of low rank symmetric positive semi-definite and non-negative diagonal matrix. Let me ...
2
votes
1answer
128 views

Prove a matrix is positive definite

Please, can somebody help me with this problem? [Ciarlet 5.3-1] Let $A$ be an invertible Hermitian matrix, with the splitting $A = M-N$, $M$ being an invertible matrix. Prove that, if the Hermitian ...
1
vote
1answer
88 views

Proof of Nonnegativity Inequality

Prove the Inequality: $$\sum_{i,j}\left ( (PAQ)_{i,j}\frac{B_{i,j}^2}{A_{i,j}}- (PBQ)_{i,j}B_{i,j}\right ) \geqslant 0$$ Given that: $P$ and $Q$ are $n$x$n$ and $m$x$m$ symmetric matrices, $A$ ...
1
vote
1answer
49 views

Prefactoring to solve many similar linear systems

I am designing an algorithm that needs to solve many (large) linear systems of the form $$\Phi^\top D_i\Phi \vec x_i=\vec r_i,$$ where $\Phi\in\mathbb{R}^{m\times n}$ with $m>n$ is fixed. We will ...
4
votes
2answers
164 views

Is Householder orthogonalization/QR practicable for non-Euclidean inner products?

The question Is there a variant of the Householder QR algorithm to orthonormalize a set of vectors with respect to an inner product if no orthonormal basis is known a priori? Background Let's ...
0
votes
2answers
113 views

square root of a symetric matrix

I have a symmetric matrix which positive-definite, but it contains zero as eigen value. So the method of Cholesky does not work, could someone give another method to do this? I do not want an ...
4
votes
1answer
226 views

constructive canonical form of orthogonal matrix

For every orthogonal matrix $Q$ over the reals there is an orthogonal matrix $P$ and a block diagonal matrix $D$ such that $D=PQP^{t}$. Each block in D is either $(1)$, $(-1)$ or a two dimensional ...
-4
votes
1answer
74 views

Can we conclude that this matrix is definite positive? [duplicate]

Let $A$ be a $n\text{-by-}m$ matrix. Suppose that columns of $A$ are linearly independent. Can we conclude that $A^TA$ is definite positive? Could you help me with proof? Thanks.
0
votes
1answer
93 views

Could someone help me to prove that this symmetric matrix is definite positive?

Let $a_{ij}\in\mathbb{R}$ for all $i,j\in\{1,...,n\}$ and $m\in\mathbb{N}$. Consider the matrix below. $$B=\begin{bmatrix} \sum_{k=1}^n(a_{1k})^2 & \sum_{k=1}^na_{1k}a_{2k} & \cdots & ...
2
votes
1answer
92 views

I would like a hint in order to prove that this matrix is positive definite

Let $a_{ij}$ be a real number for all $i,j\in\{1,...,n\}$. Consider the matrix below. $$B=\begin{bmatrix} \sum_{k=1}^n(a_{1k})^2 & \sum_{k=1}^na_{1k}a_{2k} & \cdots & ...
0
votes
1answer
2k views

Finding matrix inverse by Gaussian Elimination With Partial Pivoting

Hello guys I am writing program to compute determinant(this part i already did) and Inverse matrix with GEPP. Here problem arises since i have completely no idea how to inverse Matrix using GEPP, i ...
0
votes
1answer
79 views

Spectral radius of $A$ and convergence of $A^k$

I'm trying to understand the proof of first theorem here. Maybe it's very simple but I would like your help because I need understand this, I have no much time and my knowledge about this subject is ...
1
vote
1answer
224 views

If modulus of each one of eigenvalues of $B$ is less than $1$, then $B^k\rightarrow 0$

Let $B$ be a $n\times n$ matrix and let $X$ be the set of all eigenvalues of $B$. Prove that if $|m|<1$ then $\lim \limits_{k\rightarrow\infty}B^k=0$, where $m=\max X$. Thanks. Actually, there ...
1
vote
1answer
77 views

Represent in a matrix form: $\sum_{ijl}(F_{il}-F_{jl})^2W_{ij}$

How can I represent this in a matrix form: $\sum_{ijl}(F_{il}-F_{jl})^2W_{ij}$ where all the entries are real and $W$ is a known(constant) matrix and $F$ is a rectangular matrix. When I say matrix ...
0
votes
1answer
100 views

A an nxn matrix. P a permutation matrix that permutes columns of A. How many operations does P*A involve?

Essentially, I am supposed to count how many operations a particular computational algorithm involves, and I've gotten stuck on this one part. My understanding is that for two nxn matrices, matrix ...
7
votes
1answer
290 views

Eigenvalues of a tridiagonal trigonometric matrix

Let $A$ be the diagonal matrix w/alternating in sign diagonal entries: $$ A = \begin{pmatrix} (-1)^{n-1} \tan\left(\frac{\pi}{2n+1}\right) & 0 & 0 & \ldots & 0 \\ 0 & ...
0
votes
1answer
711 views

Prove that if $A$ is symmetric and has a LU-decomposition then $A=LDU' \Rightarrow U'=L^T$, where $L^T$

Suppose the matriz $A$ has a LU-decomposition, in other words, suppose there exists matrices $L$ and $U$ such that $A=LU$ where $L$ is lower triangular and $U$ is upper triangular. We can to prove ...
2
votes
1answer
99 views

Ways of computing $A^\infty$

As a follow-up on this question, I would like to ask which one is the better way of computing $A^\infty = \lim_{n \rightarrow \infty} A^n$. Repeatedly square, computing $A^2, A^4, A^8$ and so on. Do ...
2
votes
0answers
191 views

Proving invertibility of matrices using banachs lemma

I'm studying for finals and trying to understand how you can possibly use banach's lemma for anything worthwhile, more particularly we have a bunch of sample questions that say it can be used to prove ...