Questions on the various algorithms used in linear algebra computations (matrix computations).
5
votes
1answer
312 views
Trace minimization with constraints
For positive semi-definite matrices $A,B$, how can I find an $X$ that minimizes $\text{Trace}(AX^TBX$) under 'either' one of these constraints:
a) Sum of squares of Euclidean-distances between pairs ...
5
votes
1answer
724 views
Block inverse of symmetric matrices
Let us assume we have a symmetric $n \times n$ matrix $A$. We know the inverse of $A$.
Let us say that we now add one column and one row to $A$, in a way that the resulting matrix ($B$) is an $(n+1) ...
2
votes
2answers
164 views
Generating unitary matrices numerically - “close” to the identity element
EDIT: broke this into two parts - for these were two different questions.
For numerically obtaining the stabilities of a matricial equation, i need to generate an ensemble of matrices that are ...
1
vote
1answer
74 views
Floating point arithmetic
How can I prove that : a real number has a finite representation in the binary system if and only if it is of the form $$\pm \frac{m}{2^n}$$ where n and m are positive integers.
7
votes
1answer
110 views
Why would $(A^{\text T}A+\lambda I)^{-1}A^{\text T}$ be close to $A^{\dagger}$ when $A$ is with rank deficiency?
In many applications that is not with high requirements, it is common to use $(A^{\text T}A+\lambda I)^{-1}A^{\text T}$ or $A^{\text T}(AA^{\text T}+\lambda I)^{-1}$ ($\lambda$ is small) to ...
4
votes
6answers
598 views
A book for self-study of matrix decompositions
I am a third year math student and I noticed that there are many uses for decomposing a matrix (I mean decompositions like SVD, LU etc').
Is there a good book for self-study of the subject ?
Note ...
6
votes
4answers
168 views
Conditions for the equivalence of $\mathbf A^T \mathbf A \mathbf x = \mathbf A^T \mathbf b$ and $\mathbf A \mathbf x = \mathbf b$
I have an application where I have to minimize a cost function of the form:
$J(\mathbf x) = \| \mathbf A \mathbf x - \mathbf b \|^2 \quad (1)$
By calculating the gradient, I derived that I have to ...
2
votes
1answer
279 views
Are there any decompositions of a symmetric matrix that allow for the inversion of any submatrix?
I am given a $J \times J$ symmetric matrix $\Sigma$. For a subset of $\{1, ..., J\}$, $v$, let $\Sigma_{v}$ denote the associated square submatrix. I am in need of an efficient scheme for inverting ...
2
votes
1answer
344 views
Fourier transform over a diagonal matrix
Let $F$ be a $100 \times 100$ DFT matrix, and $U$ be a diagonal matrix with its diagonal entries being all positive, denoted by $U=\mathrm{diag}(u_1, u_2,\cdots, u_{100})$. My question is:
Under ...
9
votes
2answers
517 views
Augmented Reality Transformation Matrix Optimization
i am a software developer, i'm working on an Augmented Reality system.
I'd like to receive some advice in order to optimize my math model. My program has to be slim and fast.
Here's the situation:
...
4
votes
1answer
782 views
Linear least squares with inequality constraints
I'm trying to follow this older paper, page 19.
The goal is to solve:
$\min \|Ax-b\|^2 s.t. Gx \ge h$ given $A, G, b, h$
By combining the equations into a single LCP of the form:
$Mz + q = w$ s.t. ...
2
votes
1answer
309 views
Product of positve definite matrix and seminegative definite matrix
Let $A$ a spd (symmetric positive definite) matrix and $B$ a symmetric seminegative definite matrix. Is tr $AB \leq 0$ and more general is $AB$ seminegative definite?
I know that tr $AB \leq 0$ ...
1
vote
2answers
90 views
+100
On integral of a function over a simplex
Help w/the following general calculation and references would be appreciated.
Let $ABC$ be a triangle in the plane.
Then for any linear function of two variables $u$.
$$
\int_{\triangle}|\nabla ...
1
vote
1answer
47 views
How to find the Householder transformation?
Assume $x=(1,0,4,6,3,4)^T$. Find a Householder transformation and a positive number $\alpha$ such that $Hx=(1,\alpha,4,6,0,0)^T$.
I'm sorry that I don't know how to start with this problem. A ...
1
vote
2answers
54 views
Lanczos vectors
I am trying to implement the Lanczos algorithm. If I implement it in Fortran or C, (i.e. in finite precision), will the vectors generated at each iteration still preserve their linear independence? ...
1
vote
0answers
50 views
fixed point spectral radius
We have the following stationary matrix iteration
$$x_{k+1} = Mx_k + c$$
where $M$ is nxn matrix and $c$ is a vector.
Let $r(M)$ denote the spectral radius of $M$.
Show that spectral
radius ...
0
votes
1answer
39 views
Gaussian Elimination Triangular Factoration
Factor $$\left[\begin{array}{ccc}1&4&0\\0&1&0\\0&3&1\end{array}\right]$$ into LU (L lower triangular and U upper triangular)
0
votes
1answer
77 views
Suitable Loss function for Order preserving Factoring of a matrix?
(Old-Question)
Given a $n\times n$ symmetric matrix $X$, I would like to factor it using a vector $c$ of size $n \times 1$ such that: $\sum_{i,j} [X_{ij} \cdot c_i\cdot c_j]$ is minimum. How can I ...
0
votes
2answers
199 views
Ill-conditioned matrix [duplicate]
Possible Duplicate:
Inverse matrices are close iff matrices are close
Consider this problem:
$Ax = b$
I want to solve it/find x and the matrix A is ill-conditioned. Why is the fact "A is ...
