# Questions tagged [matrix]

Matrix is a rectangular array of elements (e.q. numbers, symbols, or expressions), arranged in columns and rows.

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### How to prove the 2-norm of an invertible matrix is exactly the reciprocal of its minimum singular value?

If a matrix $A_{n\times n}$ is invertible, then $\left\|A^{-1}\right\|_2 = \dfrac{1}{\min\limits_{i} \sigma_i}$ where $\sigma_i$ is the $i$-th singular value of $A$
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### How to find closed form $C$ such that $CC^T = AA^T + BB^T$

How to find $C$ such that $CC^T = AA^T + BB^T$, $A$ and $B$ are known. $A = \left(\begin{matrix}X\\Y\end{matrix}\right)$, $B = \left(\begin{matrix}0\\cY\end{matrix}\right)$, $c$ is a constant. To ...
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### Difference between eigendecomposition and singular value decomposition for Hermitian matrices

Let consider the following Hermitian matrix ...
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### Is there any rapid way to calculate the determinant of NXN covariance matrix?

I searched the web and found some C code for calculating the determinant of a $n\times n$ matrix. This code however seems timing complexity, and run pretty slow especially when handling a larger ...
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### Solve eigenvalue problem using finite differences without vectorization

I am interested in solving the problem $-A u = \lambda u$ on a finite differences grid on a square. In my case, the operator $A$ is of the type $-\Delta + \mu I$, where $\mu$ is there to impose some ...
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### what does “D = diag(W.1)” means?

, what does “D = diag(W.1)” means?on page #2, just below equation (6) PFA screenshot and here is the link of the paper - original paper
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### Matrix factorization empty rows and columns

I would like to do non negative mf and I wanna ask a question about my main matrix. The question is should I include rows and columns that have no non-zero entry in them. I think if there is not a ...
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### MATLAB: Matrix whose elements depend on its indicies

I am trying to put the function $$f(\mu,\nu) = i^{\nu-\mu} \sum_{0}^{19} H_{\mu-\nu}(7j) + \delta_{\mu,\nu}\ ,$$ $\mu, \nu =-3,-2,...2,3$ into a 7x7 matrix, where $H$ is the Hankel function of the ...
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### Multiplication of random sparse matrices

I am given 2 unstructured (i.e. do not posses any special sparsity pattern like banded/triangular/etc.) sparse matrices $A$ and $B$ of dimension ($n$ x $n$) and density $d$ each (thus each matrix ...
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### Evaluating a quadratic form with an inverse of a sparse PD matrix, comparison between using the inverse vs using a Cholseky decomposition

I have the following quadratic form I need to evaluate: $x^T A^{-1} y$, where $A$ is a sparse positive definite matrix, $x, y$ are sparse vectors. Now assume that I am given for free both $A^{-1}$ ...
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### Linear solve using CHLOMOD in C

I am using the open-source CHLOMOD (as here http://faculty.cse.tamu.edu/davis/suitesparse.html) in order to solve a linear system Ax=b (performing A/b=x) in my domain decomposition code but I am ...
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### Matlab Vectorization of columns in a 2D matrix and single element multiplication

MATLAB. I am trying to vectorize a loop in which each column vector of a 2D matrix (n-by-n) is found by multiplying each single element in a diagonal matrix with a column vector in another n-by-n 2D ...
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### Overcoming floating point issues when Inverse does not exist but determinant provides nonzero result

I have an expression (let's say determinant of matrix A) expressed in symbolic form in terms of 2 decision variables x1, x2 and 2 parameters q1 and q2. I'm minimizing this using fmincon for different ...
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### Gilbert-Peierls algorithm for LU Decomposition

I searched for Gilbert-Peierls algorithm, but I haven't found anything useful (well, I found this, but it's not working as it should). I think the problem is the second part, and also that those lines:...