3
votes
3answers
55 views

Index notation for inverse matrices

I have a question: There is an standard way to write the inverse of a matrix in index notation?. The reason is that I don't want to write $(A^{-1})_{ij}$ or $(A^{-1})_i^j$ or $(A^{-1})^{ij}$ using ...
2
votes
1answer
78 views

Why we use $\mathbb{R}^{m \times n}$ notation instead of $\mathbb{R}^{n \times m}$?

I just realised, that I use all the time the notation $\mathbb{R}^{n \times m}$, and all books and papers use $\mathbb{R}^{m \times n}$. $\mathbb{R}^{n \times m}$ is more sympathetic for me, because I ...
2
votes
1answer
13 views

Consider a symmetric matrix $X$ with eigendecomposition $X=UVU^T$, how to call $\sum_{v_{k,k}>0}v_{k,k}u_ku_k^T$?

Consider a symmetric matrix $X$ with eigendecomposition $X=UVU^T$ How do people call $\sum_{v_{k,k}>0}v_{k,k}u_ku_k^T$? Sum of positive components of $X$? The positive semi definite part of $X$? ...
0
votes
2answers
32 views

Scalar to the power vector

I have a formula that contains $\log (I + e^{X})$ term where $X$ is an $N$ dimensional vector ($x_0, x_1, ..., x_{N-1}$) and $I$ is the unity vector. I don't know how the exponentiation and log ...
0
votes
2answers
29 views

Let $V$ be the set of all ordered pairs of real numbers, and consider the following addition and scalar multiplication operations on $u$ and $v$

I'm confused on b) and c) , what do the terms in it mean? I'm also having trouble finding e) ..
7
votes
3answers
805 views

Meaning of math symbol ~

Segment of Example: t = ... More usefully, we have: t ~ n*log(n) Note: ~ means "similarity" like in geometry, same shape but not same size. How is it interpreted here? Edit: yes, t depends on n ...
2
votes
2answers
72 views

Dot product notation

Let $\mathbf{A=(a_1,a_2,\ldots, a_n)}$ and $\mathbf{B=(b_1,b_2,\ldots,b_n)}$. Many linear algebra books and texts define the dot product as $$ \mathbf{A\cdot B^T=a_1b_1+a_2b_2+\cdots+a_nb_n} $$ where ...
2
votes
2answers
55 views

Both Linearly Independent and Dependent?

Is it possible for two vector functions of, for the moment's simplicity, one variable be both independent and dependent? The reason I'm asking this is because on a problem from a book of mine (not ...
1
vote
0answers
22 views

element subset for adjacency matrix

I am trying to create an element of a matrix that is a subset of a larger matrix. However, I am told that my subscripts do not match. I wanted other people's opinion as to what I am doing wrong and ...
2
votes
2answers
57 views

Linear algebra notation involving $\preceq$

I am trying to read a paper which involves some linear algebra. I would appreciate if anyone could clarify what the below statements exactly mean. 1-) $0 \preceq B^TB \preceq A^TA$ where $A \in ...
2
votes
1answer
19 views

$[R(T)]^o$ $?!?!$

I was studying linear algebra(Linear Transformations) a day back and came across this notation and couldn't understand what it meant. Is it the $interior$ of the $Range$ of linear transformation $T$? ...
0
votes
0answers
115 views

What does a dot in a circle mean?

I'm looking at some formulas involving matrices (in the context of machine learning, but I'm not sure it's relevant) and I came across $\odot$. What could this mean? The context is $M \odot N$, where ...
0
votes
1answer
71 views

Notation for dimension of vector space

Is it an unusual notation to write $|V|$ for the dimension of a vector space $V$? Is it ok to use it if you blur the distinction between the grid for the finite element method and its associated ...
1
vote
1answer
39 views

matrix inverse in tensor notation

Suppose there is a matrix $A$ that transforms vectors, $$ Y = A x $$ Now express this in some other coordinate system, with $x = B z, \,\, y = B w$, so \begin{align*} & Bw = A B z \\ ...
0
votes
0answers
54 views

Notation in Linear Algebra

What does $(A\mid b)$ denote in Linear Algebra? Specifically in the context of the following question: "If $(A\mid b)$ is in reduced row echelon form, prove that A is also in reduced row echelon ...
3
votes
1answer
50 views

Is there a way in matrix math notation to show the 'flip up-down', and 'flip left-right' of a matrix?

Title says it all - is there an accepted mathematical way in matrix notation to show those operations on a matrix? Thanks.
2
votes
2answers
57 views

Notation for vector space dimension and usage of $\times$ operator

I would like to describe a real vector space with dimension $a \times b$ (as in $a$ times $b$). Is it correct to describe it with $$ \mathbb R^{a \times b},$$ or would that imply the space of matrices ...
0
votes
2answers
38 views

How would I express the statement “Let H be a subspace of V” in mathematical notation?

How would I express the statement "Let H be a subspace of V" in mathematical notation? Does something like this work? $$ ( \ \ H(\mathbb{R})\subset V(\mathbb{R}) \ ) $$
3
votes
2answers
47 views

Notation - Transpose of Block Matrices [Lay P121 Q2.4.12]

Definition of Transpose is $(A^T)_{ij} = A_{ji}$ $1.$ Why $\begin{bmatrix} M & N \end{bmatrix}^T = \begin{bmatrix} M^T \\ N^T \end{bmatrix}$, and NOT $\begin{bmatrix} M \\ N\end{bmatrix}$? ...
0
votes
0answers
41 views

Is there a traditional name for the “eigenspace” function?

Let $A$ denote a field, $X$ denote an $A$-vector spaces, and suppose $\varphi : X \rightarrow X$ is a linear transformation. Is there a traditional name for the corresponding "eigenspace" function? By ...
2
votes
1answer
54 views

change of basis and inverse in tensor notation

I'm trying to study tensors from several textbooks. I would like it if someone could confirm my understanding of a particular easy example. It is from Itskov, Tensor Algebra and Tensor Analysis for ...
2
votes
4answers
400 views

Matrix Mathematical Notation

I am trying to work out the mathematical notation for combining the columns of two matrices, $$A=\begin{pmatrix}1 & 3 & 5 \\ 2 & 4 & 1 \\ 3 & 7 & 9\end{pmatrix}$$ and ...
0
votes
2answers
46 views

Which is the right notation to write the solutions of a single equation with two variables?

I've recently encounter the problem of solving these two equations in $\mathbb{N}$ and $\mathbb{Z}$ for answer a question in SE. $$ax-by=0$$ and$$ax+by=0$$ (Please note: are two single equations ...
3
votes
1answer
59 views

Notation of set of vectors

I was wondering if someone could clarify this for me. In my lecture notes my lecturer keeps writing "Let $\{\textbf{v}_1,\ldots, \textbf{v}_n\}\in \mathbb{R}^m$ be a collection of vectors $\dots$" ...
0
votes
2answers
35 views

Notation question: defining a matrix as $P=\sum_{i=1}^{k}v_{i}v_{i}^{T}$

I've seen in a paper the following sentence: Let $V$ be a $k$-dimensional subspace of $\mathbb R ^d$ and let $v_1 ,...,v_k$ be an orthonormal basis of $V$. Define ...
1
vote
1answer
37 views

meaning of subscript notation $\ 1_{a=a'}$

I'm not familiar with the meaning of the 1 with the subscript notation $\ 1_{a=a'}$ and $\ 1_{b=b'}$, where (a,a') and (b,b') are simply coordinates of a matrix. Could anyone explain this to me ...
0
votes
0answers
27 views

Notation from nowhere - $V(T_A)$

This is a poor question I have. In an exam question, $T_A$ is a linear map. What is $V(T_A)$ ? It is not stated in the question. And I have not seen it before. Maybe some help to figure it out, he ...
0
votes
3answers
75 views

Function Notation question that needs an answer

$f(x)= f(x+1)+3$ and $f(2)= 5$, determine the value of $f(8)$. I don't understand how $f(x)$ can equal $f(x+1)+3$
0
votes
0answers
11 views

Equation form representation

In an equation, when I assume that a variable is given in a specific unit (e.g. ms), I write “x (ms)”. For example, consider the following equation: $$Y(ms)=2*z+x(ms)$$ Now, if I give x in ms, Y ...
2
votes
1answer
52 views

Notation Question for summations

I came across this notation in a textbook of Algebra. With respect to the definition of linear independence in a Vector Space $V$. We define a subset $S = \{\alpha_i \ | \ i \in I\} \subset V$ as ...
1
vote
2answers
42 views

Confusion regarding notation on a matrix which have $I$ as an element

I'm given this transformation matrix for a linear map between two general vector spaces. I've never seen this notation before: $\begin{bmatrix} I &0 \\ 0& 0 \end{bmatrix}$, where I is the ...
0
votes
1answer
55 views

confusion about Einstein notation

I have the following expression $$ \partial_k\sigma_{ik} = \partial_i\partial_ku_k + \partial_k^2u_i $$ I know that a repeated index means summation. So for a 2D space $(x, y)$ I would get (for the ...
0
votes
1answer
65 views

What does this notation mean? $[p]^B$

The question I'm trying to answer is: Find an ordered basis $B$ of $\mathbb C_3[x]$, such that $$[p]^B= \begin{pmatrix}1\\0\\0\\i\end{pmatrix}$$ for the polynomial $p=2+2x+2x^2+2x^3$. The notation ...
2
votes
3answers
64 views

Find the general solution of the system

Given: \begin{cases} x- z = 1 \\ y + 2z - w = 3 \\ x + 2y + 3z - w = 7 \end{cases} My work: Use Gauss method Leading variables: x,y, and z Free variables: w Express the leading variables x,y,and z ...
1
vote
2answers
51 views

Notation for Subspaces

Is there a proper notation for denoting subspaces? For example, if $U$ is a subspace of some vector space $V$. I would usually just write "the subspace $U \subseteq V$" but I'm wondering is there is a ...
1
vote
3answers
59 views

What does $f|_V$ mean?

I want to proof: If $W$ is vector space, $U, V \subseteq W$ linear subspaces, and $f : W \rightarrow Z$ is homomorphism, $\ker f \subseteq U$ and $W = U \oplus V$ then $f|_V : V \rightarrow Im(f|_V)$ ...
0
votes
0answers
75 views

Notation for Kronecker product of a matrix and itself?

What is the notation for the Kronecker product of a matrix and itself? In other words, is there a short-hand way I can express the following: $X⊗X$ $X⊗X⊗X$ $X⊗X⊗X⊗X$ Where $X$ is a matrix? What ...
0
votes
2answers
59 views

What does power of '+' in $\hat{\beta}_j=\operatorname {sign} (\hat{\beta^0_j})(|\hat{\beta^0_j}|-\gamma)^+$means?

I encountered the following equation in a paper. $$\hat{\beta}_j=\operatorname {sign} (\hat{\beta^0_j})(|\hat{\beta^0_j}|-\gamma)^+$$ What does the power of '+' mean? The paper can be viewed at: ...
2
votes
3answers
186 views

Basic question: what does the notation $[A,B]$ mean?

If $A$ and $B$ are both matrices, what is $[A,B]$? I understand that it is a commutator and that $[A,B]=AB-BA$, but since I don't know what a commutator is, none of this information is telling me ...
1
vote
1answer
63 views

Can the elements of a direct sum be thought of like that?

I've asked here about the tensor algebra, and I think that my problem is being able to realise the elements of a direct sum as linear combinations. Indeed the rigorous definition I have of the direct ...
0
votes
3answers
54 views

Where V is a Vector Space, $\forall \overrightarrow{v} \in V, 0\overrightarrow{v} = \overrightarrow{0}$

I'm denoting all vectors as such: $\overrightarrow{v}$. Any variable without an arrow above is a scalar. Suppose $V$ is a vector space over $F$, with additive identities $\overrightarrow{0}$ and $0$ ...
2
votes
3answers
265 views

Notation: subscript vs. superscript for coordinate vector fields

Some books write the coordinate vector fields with a subscript as $$\frac{\partial}{\partial x_i}$$ while some write it with a superscript as $$\frac{\partial}{\partial x^i}.$$ Is there a conceptual ...
2
votes
1answer
85 views

Why is $m$ used as the variable for slope in slope-intercept form?

I was wondering if you could answer a question I have on slope intercept form of a linear equation. I know its $y=mx+b$, but why is it $mx+b$? Don't get me wrong. I know that $m$ is the slope and $b$ ...
0
votes
0answers
25 views

Notation for the singular value decomposition.

Consider the singular value decomposition of a Hermitian $\mathbf{M}$ of the form $\mathbf{M} = \mathbf{U}\boldsymbol{\Lambda}\mathbf{U}^{\dagger}$. Then, for the SVD of another Hermitian ...
0
votes
3answers
85 views

What do exponents of $\mathbb{R}$ denote?

An equation in my machine learning class says $$ x= \begin{bmatrix} x_0 \\ x_1 \\ \vdots \\ x_n \\ \end{bmatrix} \in ℝ^{n+1} $$ I'm reading this as "x ...
1
vote
0answers
39 views

Notation for linear transform where only a subspace of the domain is used

I posed this question (sort of) here: A 2x2 matrix $M$ exists. Suppose $M^3=0$ show that (I want proof) $M^2=0$ Suppose I have a linear transformation over a field $K$ $T:K^3\rightarrow K^3$ ...
4
votes
4answers
619 views

A 2x2 matrix $M$ exists. Suppose $M^3=0$ show that (I want proof) $M^2=0$

I'm sure I've done this before in abstract algebra. Regardless it's escaped me now. I have proved that for $T:U\rightarrow V$, with $dim(U)=m$ and $dim(V)=n$ that $rank(T)\le m$ which is obvious, ...
0
votes
1answer
98 views

What is the dot product between a vector of matrices?

There is a notation used in many sources (e.g. Wikipedia: http://en.wikipedia.org/wiki/Exponential_family) for the natural parameters of exponential family distributions which I do not understand, and ...
0
votes
1answer
60 views

Notation question in linear algebra problem

I am confused by the symbol $\mathcal{L}$. What do $\mathcal{L}(V)$ and (especially) $\mathcal{L}(\mathcal{L}(V),\mathcal{L}(W))$ in b) mean? Let $T:V\to W$ be an isomorphism. For each ...
2
votes
1answer
32 views

What is the notation $M_{n}(\mathbb{R})$?

I'm familiar with $M_{m\times n}(\mathbb{R})$ being the set of all $m\times n$ matrices, but I'm not sure I know what this one is.