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Results tagged with matrices
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user 88133
Questions where the notion matrix has an important or crucial role (for the latter, note the tag matrix-theory for potential use). Matrices appear in various parts of mathematics, and this tag is typically combined with other tags to make the general subject clear, such as an appropriate top-level tag ra.rings-and-algebras, co.combinatorics, etc. and other tags that might be applicable. There are also several more specialized tags concerning matrices.
15
votes
Accepted
Product of conjugate matrices in $\mathrm{SL}(2, \mathbb{Z})$
See Keith Conrad's notes http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/SL(2,Z).pdf, particularly Example 2.5. Let us write (as Conrad does) $S = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ …
- 6,237
6
votes
Accepted
What is the minimum number of multiplications for $2\times 3$ and $3\times 2$ multiplication?
Instead of a map that takes matrices $A$ and $B$ and tries to find the product $AB$, consider a map that takes three matrices $A,B,C$ and returns "the coefficient of $C$ in $AB$", or more precisely the … With all of this, the original question is about multiplication of $2 \times k$ by $k \times 2$ matrices. …
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2
votes
The functional equation $T(x\otimes y)=T(x)\otimes T(y)$ on the matrix algebra
What is a linear operator---do you require $T_n(X_1 X_2) = T_n(X_1) T_n(X_2)$? If not, $T_n(X) = c^n X$ for a fixed scalar $c$ works. Even if yes, I think that $T_n(X) = X$ if $n$ odd, $0$ if $n$ even …
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11
votes
0
answers
252
views
Poset of nonvanishing minors of a matrix
What if we restrict to integer $0$-$1$ matrices?
For instance, $P$ can't be a linear order, except in the pretty trivial case that $M$ has at most one nonzero entry. …
- 6,237
1
vote
Permanent of distorted matrix
Say $M$ is $n \times n$. For $1 \leq k \leq n$, let $P_k(M)$ be the sum of all of the permanents of $k \times k$ minors of $M$; and let $P_0(M) = 1$. For example $P_n(M)$ is the permanent of $M$, whic …
- 6,237
1
vote
Existence of polynomial equation system solution
It's not quite a general $(n-1)$-plane because it's spanned by rank one matrices. But it still misses $N$ anyway. … The family of $(n-1)$-planes spanned by rank one matrices is irreducible, so it's sufficient to show a single one that misses $N$. …
- 6,237
1
vote
The rank of the Hadamard product
Say $E$ contains a nonzero transversal (I welcome any suggestions of better terminology) if there is a collection of entries of $E$ with the properties (1) no two of the entries are in the same row or …
- 6,237
2
votes
Accepted
Inverse of a larger matrix where the inverse of the submatrix is known
You know $\begin{pmatrix} A & 0 \\ 0 & 1 \end{pmatrix}^{-1} = \begin{pmatrix} A^{-1} & 0 \\ 0 & 1 \end{pmatrix}$, and from there you can make two successive rank-$1$ modifications, first adding $b$ al …
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1
vote
About local maxima of multivariable polynomials
The set of critical points (in the domain) of a polynomial is the solution set of a system of polynomial equations viz the vanishing of the first derivatives. So it has finitely many irreducible compo …
- 6,237
1
vote
Accepted
Rank of matrices and secant varieties
The elements of $\mathbb{P}^N$ are $(n+1) \times (n+1)$ matrices (up to a scalar factor) and rank with respect to $S$ is ordinary matrix rank. …
- 6,237