All Questions
Tagged with linear-algebra terminology
34 questions
0
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87
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Is there a name for "applying linear operations to vector sequences from the right"?
Let $v_1,...,v_n\in\Bbb R^d$ be a sequence of vectors. When we say that we "linearly transform" this sequence, we mean that we apply a linear transformation $T\in\Bbb R^{d\times d}$ to each ...
- 13.6k
6
votes
2
answers
647
views
Is there a name for matrices of the form $a_{ij}=\frac{1}{a_{ji}}$?
I have a matrix that is “kind of symmetric.” Specifically, it is an $n \times n$ real matrix such that the entries $a_{ij}=1/a_{ji}$ whenever $j \ne i$. I want to investigate the properties of this ...
- 79
5
votes
2
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508
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Is there a name for this family of matrices?
Let $0<a_1<a_2<\cdots<a_n$ and let $A$ be the symmetric $n\times n$ matrix with
${ij}^\text{th}$ entry $A_{ij}=\min\{a_i,a_j\}$.
For example, if $a_i=i$ for each $i\le n=5$ then
$$A=\begin{...
- 51
2
votes
1
answer
131
views
Name for a sum of dyadic vector products
Question:
is there a name for the following operation
$$\sum_{i=1}^n\sum_{j=1}^mx_iy_j^T,\ x_i,y_j\in \mathbb{R}^k$$ i.e. for generating a square matrix that is the sum of the cartesian product of a ...
- 13.2k
2
votes
0
answers
85
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Name for closure property: set of maps closed under taking $(f,g)\mapsto (f-g)/2$
Suppose that $F$ is a collection of functions mapping some set $\Omega$ to $\mathbb{R}$, with the following closure property: whenever $f,g\in F$, we also have $(f-g)/2\in F$. Is there a name for this ...
- 6,541
1
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0
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216
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Is there a name for the algebraic structure of all real matrices?
I know there are matrix rings, but what do we call the structure of the set of ALL real matrices, with the usual sum and product? We are looking for a kind of algebra whose operations aren't defined ...
- 65
3
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2
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569
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What do you call a scaled orthogonal map?
What do you call a linear map of the form $\alpha X$, where $\alpha\in\Bbb R$ and $X\in\mathrm O(V)$ is an orthogonal map ($V$ being some linear space with inner product)? Are there established names, ...
- 13.6k
0
votes
2
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1k
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Is there a term for the operation of multiplying the product of two matrices by the transpose of the first matrix? [closed]
Is there a term for the operation $A B A^T$? In colloquial terms, I might call this a "sandwich" of a matrix between another matrix and the transpose of that other matrix.
How about for the ...
- 111
2
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0
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199
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What is the name of this tensor?
A matrix M is usually called a hollow matrix if all of its diagonal elements are zero:
$$ M_{pp} = 0, \quad \forall \: p.
$$
We can generalize this to an $n$-way tensor T, such that:
$$ T_{p_1 \cdots ...
- 47
3
votes
1
answer
844
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Origin of the term relaxation method in numerical analysis for iteratively solving linear equations
In the iterative methods for solving a system of linear equations, a term called relaxation method is often appears along with Jacobi and Gauss Seidel methods. As per the Earliest Known Uses website,
...
- 879
2
votes
2
answers
107
views
Solution to a matrix optimisation problem with a particular structure
Does a matrix of the form $A_{ij} = v_i + v_j$ for some arbitrary vector $v$ have a particular name?
I am attempting to find the closed form solution (if it exists, although it looks like it might) ...
- 31
2
votes
0
answers
342
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Is there a name for this "inner product" on projective space?
$\newcommand{\proj}{\mathbb{P}}\newcommand{\complex}{\mathbb{C}}\newcommand{\ip}[2]{\langle #1 , #2\rangle}\newcommand{\abs}[1]{\lvert #1 \rvert}$There is a natural bijection $\phi: \proj(\complex^n)\...
- 980
6
votes
1
answer
259
views
Name for a matrices having a specific property
is there an established name for the property that a square matrix can be made symmetric by permutation of its columns?
Is it possible to recognize those kind of matrices efficiently?
- 13.2k
0
votes
1
answer
145
views
Is there a name for $f(M, x) = x^\top M x$? [closed]
I often encounter things of the form $x^\top M x$, where $M$ is symmetric positive (semi-)definite. Is there a term for that? I know related terms:
We can say $M$ is a bilinear form, $M(x,y) = x^\top ...
- 127
2
votes
3
answers
420
views
A function in $\mathbb{R}^n$ is equal to its linearization in each point
I have a function $P: \mathbb{R}^n \to \mathbb{R}^n$. This function satisfies:
$$ P(\vec{x}) = J_P(\vec{x}) \cdot \vec{x}$$
where $\vec{x}\in \mathbb{R}^n$, $J_P$ is the Jacobian of $P$ and "$\cdot$" ...
- 469
5
votes
1
answer
571
views
Can I assign the term "is eigenvector" and "is eigenmatrix" of matrix $P$ in my specific (infinite-size) case?
Remark: I asked this in MSE, the question got views and votes but seemingly no one had an answer so far.
Background: I'm rereading a couple of my exploratory (surely not research-level) math-essays ...
- 5,342
3
votes
0
answers
434
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Name for matrices with vanishing row and column sums
Question:
is there a special name for matrices whose rows and columns sum to zero?
I actually need information about those matrices and thus a keyword for online search.
Edit:
as there apparently is ...
- 13.2k
5
votes
1
answer
515
views
Do matrices with only elements along the main and anti-diagonals have a name?
To expand upon the title, I am wondering if there is a specific name for square matrices of the form: $$M = \begin{bmatrix} a_{11} & 0 & \cdots & 0 & \cdots & & 0 & b_{1n} ...
- 169
4
votes
1
answer
4k
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Why is the matrix of all 1's called "J"? [closed]
I've seen J referenced recently in some discussion about algebraic combinatorics, and it took me a while to figure out it was the matrix of all ones. It came up without definition, and I spent too ...
- 73
10
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2
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2k
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Is there a standard name for (non-square) matrices with orthonormal columns?
One encounters often in numerics non-square matrices with orthonormal columns, i.e., $U\in\mathbb{R}^{m\times n}$, with $m > n$, such that $U^TU=I$ (but, clearly, $UU^T \neq I$).
Is there a name ...
- 20.2k
2
votes
0
answers
29
views
Terminology question- Antihermitian elements
Let $V$ be a vector-space over a field $F$, and let $B$ be a non-degenerate bilinear form on $V$.
Question: Is there a common term to call an operator $x\in End(V)$ satisfying $$B(xu,v)+B(u,xv)=0\...
- 993
7
votes
1
answer
464
views
"Unimodality" of the positive eigenvector of a non-negative irreducible matrix?
Consider an eigenvalue / eigenvector problem for a matrix $A$ that is known to be non-negative and irreducible (so the Perron-Frobenius theorem applies):
$$\sum_j A_{ij} x_j = \lambda x_i$$
Here $\...
- 884
0
votes
1
answer
317
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Some questions related to the unitary operators
A unitary operator is a surjective linear operator between complex inner product spaces, which preserves the inner product.
What is the name of the analogue for the real case? Orthogonal operator ...
- 5,529
9
votes
3
answers
390
views
Is there a standard name for the following type of linear operator?
Is there a standard name for a linear operator $T$ on a finite dimensional vector space satisfying $T^n=T^{n+1}$ for some $n\geq 1$ or, equivalently, $T$ is a similar to a direct sum of a nilpotent ...
- 38.6k
2
votes
0
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130
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Notions of singularity for symmetric bilinear maps
Given a symmetric, bilinear map $B : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^m$, a common notion of nondegeneracy is the following:
$$
\mbox{If } B(x,y) = 0 \mbox{ for all } y, \mbox{then } x=...
- 193
10
votes
0
answers
477
views
Name for an operation on matrices?
Given two matrices $A$ and $B$ of size $a \times n$ and $b \times m$ consider the following operation $A \dagger B$ whose result is an $a b^n \times n m$ matrix. $A \dagger B$ is a block matrix with $...
10
votes
3
answers
2k
views
Partial inverse of a matrix - or does it have its own name?
In my calculations I need to use something which is "between" a matrix and its inverse. That is, I invert only some dimensions. I am interested if it has an established name.
That is, a matrix (here ...
- 1,612
0
votes
1
answer
261
views
Name for a Specific Type of Non-Symmetric Bilinear Form
Let $V$ be a finite dimensional vector space, with some choice of basis $\{e_i\}_{i \in I}$. With respect to an idempotent bijection $B:I \to I$, define a bilinear form by
$$
g = \sum_{i=1}^N \lambda_{...
2
votes
2
answers
421
views
On matrix norms
It is standard to define an induced matrix norm $|||\cdot|||$ from a vector norm $||\cdot||$ in this way:
$|||A|||=\max_{x \neq 0}{\frac{||Ax||}{||x||}}$.
Suppose we define a different function of ...
- 6,962
1
vote
1
answer
206
views
What is such an equation called?
Is there a name and common technique for such equations, where $A$ and $B$ are matrices and $x$ a vector?
$Ax+f(\lambda)Bx=g(\lambda)x$.
- 6,962
1
vote
1
answer
212
views
name for a matrix operation
If $A$ is a matrix and $D$ is a diagonal matrix, is there some special name for $DAD$?
- 6,962
4
votes
1
answer
298
views
Is there a standard name for the intersection of all maximal linearly independent subsets of a given set in a vector space?
The title more or less says it all.... Let $V$ be a vector space (over your favorite field; $V$ not necessarily finite dimensional), and let $S$ be a subset of $V$. A maximal linearly independent ...
- 12.8k
3
votes
1
answer
456
views
Standard name for basis-independent submatrices?
Given a linear map $T:H\to H$ on an inner-product space $H$ and a subspace $K\subseteq H$, define the map $T_K = \pi_K T \pi_K^* :K \to K$, where $\pi_K:H\to K$ is the orthogonal projection.
As an ...
- 11.4k
2
votes
2
answers
356
views
Is there a specific name for matrices with nonsingular principal submatrices?
Is there a specific name for matrices with nonsingular principal submatrices?
- 1,638