# Questions tagged [linear-algebra]

Linear algebra deals with vector spaces and linear transformations.

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### Find linear combination with small support

Let $v_1,\dots,v_n$ be a basis of a vector subspace of $\Bbbk^N$, say for $\Bbbk$ a finite field. I would like an algorithm to find a linear combination of the $v_i$'s with small support, i.e. with ...
96 views

### Algebra in complexity theory

Recently an idea came to my mind. Suppose $V$ is vector space and $\dim V = n$. Then, since $V \simeq \mathbb{R}^n$, any conjunction of $n$ boolean formulas $\phi_1, \ldots, \phi_n$ about vectors from ...
49 views

### Approximate Matrix Multiplication with approximation guarantees that ignore large elements?

Approximate matrix multiplication is a technique to replace a matrix product $A^t B$ with a smaller product $(\Pi A)^t(\Pi B)$. Intuitively, if $\Pi$ is chosen from a suitable distribution that has ...
54 views

### Impact HHL caveat relaxation on quantum advantage

We know that there are four caveats for the exponential speedup proven for the HHL algorithm. Could anyone answer how that exponential speedup evolves as we relax the caveats? For example, the ...
148 views

### On the plausability of quantum RAM

I'm fairly new to quantum computation and quantum complexity theory, but I came across some articles that suggest that quantum RAM (QRAM) is not very realistic assumption. For example some works show ...
1 vote
63 views

### Circuit depth of linear algebra operations

I was checking the following paper  about low-depth PRFs from lattices. In table 1 on page 4, there is comparison with other constructions, and it shows evaluation depths of certain PRFs. I'm not ...
26 views

### Linear modular equalities with $0/1$ solution

Let $Ax\equiv b\bmod q$ be a $n\times n$ modular linear system known to have $0/1$ solution where $q$ is a large prime. We can solve in $NC^2$ for general linear systems using determinant and matrix ...
6k views

### Evidence that matrix multiplication is not in $O(n^2\log^kn)$ time

It is commonly believed that for all $\epsilon > 0$, it is possible to multiply two $n \times n$ matrices in $O(n^{2 + \epsilon})$ time. Some discussion is here. I have asked some people who are ...
21k views

### Complexity of Finding the Eigendecomposition of a Matrix

My question is simple: What is the worst-case running time of the best known algorithm for computing an eigendecomposition of an $n \times n$ matrix? Does eigendecomposition reduce to matrix ...