Tagged Questions
0
votes
1answer
25 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
28 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
21 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
41 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
72 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
30 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
29 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
33 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
92 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
87 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
39 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
69 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
74 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
93 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
58 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.
