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Results tagged with linear-algebra
Search options questions only
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user 6094
Questions about the properties of vector spaces and linear transformations, including linear systems in general.
13
votes
1
answer
882
views
Probability that random nonnegative integer matrix is singular
Q. What is the probability that an $n \times n$ matrix, whose elements
are independent uniformly random integers in $\{0,1,\ldots,k\}$, is singular?
For example, for $n=3$ and $k=2$, the first …
- 151k
10
votes
2
answers
1k
views
Does the hyperdeterminant calculate a quantity akin to the volume of a parallelepiped?
If $M$ is an $n \times n$ matrix, $|\det(M)|$ is the volume of the $n$-dimensional
parallelepiped spanned by the column vectors of $M$.
(Image from Wikipedia's Determinant ar …
- 151k
109
votes
15
answers
12k
views
Why are matrices ubiquitous but hypermatrices rare?
I am puzzled by the amazing utility and therefore ubiquity of
two-dimensional matrices in comparison to the relative
paucity of multidimensional arrays of numbers, hypermatrices.
Of course multidimens …
- 151k
16
votes
4
answers
914
views
Integer matrices whose determinant equals their norm
Let $M$ be an $2 \times 2$ matrix, with all entries in $\mathbb{N}$:
$$
M=
\begin{bmatrix}
a & b \\
c & d
\end{bmatrix} \;.
$$
So
$$
\mathrm{det}(M) = a d - b c \; .
$$
The
Euclidean norm
(a.k.a. F …
- 151k
16
votes
1
answer
765
views
Minimizing the excursion of a sum of unit vectors
I have $n$ unit-length vectors $v_i$ in $\mathbb{R}^3$, whose
sum is zero:
$$ v_1 + v_2 + \cdots + v_n = 0 \; .$$
Now I form the closed polygon $P$ in space by placing them head to tail.
So the vertic …
- 151k