# Questions tagged [linear-algebra]

Questions on the algorithmic/computational aspects of linear algebra, including the solution of linear systems, least squares problems, eigenproblems, and other such matters.

1,053 questions
Filter by
Sorted by
Tagged with
197 views

### What makes a good preconditioner when only a few approximate iterations are needed?

For deterministic solver of $Xw=y$, one recommendation is to pick $P$ such that $P^{-1}X$ has a low condition number. However, this condition only really matters when you want to reduce initial error ...
• 1,442
735 views

### Why does this preconditioner effectively reduce the condition number of a random SPD matrix?

Consider some randomly generated matrix $B\in\mathbb{R}^{100\times100}$ and let $A:=BB^{\top}$ On MATLAB I computed the condition number of $A$, I obtained a value of $2.8377\mathrm{e}+04$ However if ...
1 vote
63 views

### Does cblas_dgemm mutate my input matrices?

I have written a matrix class Matrix<T> for which I have implemented a wrapper function for cblas_dgemm. ...
• 11
112 views

### Global minimum of a function involving matrix exponentials

I have a matrix system $$\vec{x}(t) = e^{(iAt)} \vec{x}(0)$$ that comes from the solution of the differential system \frac{d}{dt}\vec{x} = iA\vec{x}, \end{...
48 views

• 956
201 views

### Which preconditioners make Richardson iteration convergent?

Suppose we solve an $m\times n$ full-rank system of equations $Ax=b$ by iterating the following for a small enough $\mu>0$ $$x=x+\mu B(b-Ax)$$ Is there a nice description of kinds of $B$ which make ...
• 1,442
138 views

### Column-normalized inverse?

Suppose we define $A^{*}$ of positive definite $A=X'X$ using following two steps: let $B=A^{-1}$ scale columns of $B$ to obtain a matrix with $1$'s on the diagonal For the case of singular $A$, we ...
• 1,442
61 views

### Choice between using Moore-Penrose inverse and G2 inverse

Moore-Penrose inverse for an arbitrary matrix $X\in \mathbb{R}^{n \times p}$ is defined by a matrix $X^\dagger$ satisfying all of the Moore-Penrose conditions, namely \begin{align} (1) \;\;\;& XX^\...
• 173
114 views

### Numerical representation of linear spaces

A linear space in mathematics is a set whose elements (vectors) have operations of addition and multiplication by a scalar defined in such a way that certain properties are satisfied (commutativity, ...
• 2,088
2k views

### Is there an algorithm or graph theory that allows me to not need to store an intermediate matrix when calculating AT*Y1*A + BT*Y2*B?

I have a system of conductors for which there are two dense matrices of the (complex) mutual admittances, $Y_A$ and $Y_B$, which are symmetric. Then, an equivalent nodal admittance matrix $Y_N$ is ...