Tagged Questions
0
votes
1answer
26 views
Best way to write the characteristic polynomial
In linear algebra I've seen that there are two major different ways
to write the characteristic polynomial of a mapping $f$: As
$$
...
3
votes
1answer
45 views
Geometric intuition behind the Uniform Boundedness Principle
Is there a way to visualize why the Uniform Boundedness Principle should be true? I understand the statement of the theorem but I'm having a hard time seeing a picture of it in my head.
1
vote
0answers
47 views
What to take from representation of $S_d$?
I am reading about group representations, and books I read all contain the representation theory for symmetric groups $S_d$. However none of them presents the material in a friendly way. After reading ...
1
vote
1answer
37 views
Intuition behind symmetric and antisymmetric tensors
I've been studying multilinear algebra on Kostrikin's "Linear Algebra and Geometry" and he says the following. If $V$ is a linear space, $T^q_0(V)=V^{\otimes q}$ and if $f_\sigma :T^{q}_0(V)\to ...
13
votes
1answer
117 views
What is duality?
I have seen some examples of duality. Sometimes applied to theorems, as for example Desargues theorem and Pappus theorem. Sometimes applied to spaces, for example the dual space of a vector space. ...
0
votes
1answer
73 views
Can anyone comprehend fourth-dimensional space and higher visually? [closed]
When I look at the "Clifford torus," for example, it just looks like a three dimensional object that's morphing/changing shape as it moves. Can anyone actually comprehend fourth-dimensional objects ...
1
vote
0answers
86 views
Why Markov matrices always have 1 as an eigenvalue
Also called stochastic matrix. Let
$A=[a_{ij}]$ - matrix over $\mathbb{R}$
$0\le a_{ij} \le 1 \forall i,j$
$\sum_{j}a_{ij}=1 \forall i$
i.e the sum along each column of $A$ is 1. I ...
1
vote
1answer
72 views
dim$(V)$ = $n$, dim$(W)$ = $m$ $\implies$ dim($L(V,W)$) = $nm$
I am reading Hoffman & Kunze's chapter on linear transformations, with a view towards understanding dual spaces. (I primarily want to read Calculus on Manifolds; in the first chapter of that book, ...
2
votes
1answer
61 views
Graphically, what is positive semidefinite-ness?
Suppose that we are trying to minimize a function $f$ on $\mathbb{R}^n$ and we apply Newton's method, updating:
\begin{align}
\mathbf{x}_{n+1} = \mathbf{x}_n - [\nabla^2 f(\mathbf{x}_n)]^{-1} \nabla ...
4
votes
0answers
90 views
Geometric intuition for Jordan normal forms (invariant subspaces, shearing, scaling, etc.)
I'm trying to visualize what a linear operator does to a vector space if that operator can be put into Jordan normal form.
For concrete motivation, let's take $V = \mathbb{R}^3$, with some linear ...
9
votes
4answers
198 views
Looking for an intuitive explanation why the row rank is equal to the column rank for a matrix
I am looking for an intuitive explanation as to why/how row rank of a matrix = column rank. I've read the proof at http://en.wikipedia.org/wiki/Rank_of_a_linear_transformation and I understand the ...
11
votes
3answers
129 views
Are matrices best understood as linear maps?
Any linear map between finite-dimensional vector spaces may be represented by a matrix, and conversely. Matrix-matrix multiplication corresponds to map composition, and matrix-vector multiplication ...
3
votes
3answers
146 views
What is the intuitive meaning of the basis of a vector space and the span?
The formal definition of basis is:
A basis of a vector space $V$ is defined as a subset $v_1, v_2, . . . , v_n$ of vectors in that are linearly independent and span vector space $V$.
The ...
19
votes
5answers
386 views
Motivation for spectral graph theory.
Why do we care about eigenvalues of graphs?
There must be some reason. There is an entire mathematical discipline about them.
I always assumed that spectral graph theory is an extension of graph ...
2
votes
3answers
85 views
Intuition about Hyperplane
I'm having a hard time understanding hyperplane ideas. So, can anyone explain to me how to easily understand what a Hyperplane is ?
2
votes
1answer
105 views
Intuitive interpretation of the adjacency matrix as a linear operator.
Naturally we can describe graphs via tables of "yes there is an edge" or "no there is not" between each pair of vertices, so the definition of an adjacency matrix is easily understood. Thinking of ...
1
vote
2answers
146 views
What is a intuitive proof of multivariable changing of variables formula (jacobian) without using mapping and/or measure theory?
What is a intuitive proof of multivariable changing of variables formula (jacobian) without using mapping and/or measure theory?
I was thinking that textbooks make the proofs over complicate.
If ...
9
votes
3answers
523 views
What's the Clifford algebra?
I'm reading a book on Clifford algebra for physicists. I don't quite understand it conceptually even if I can do most algebraic manipulations. Can some-one teach me what the Clifford algebra really ...
2
votes
1answer
116 views
What is the intuitive meaning of the adjugate matrix?
The definition of the adjugate matrix is easy to understand, but I have never seen it used for anything.
What is the intuitive meaning of this matrix?
Are there examples of applications which may ...
3
votes
5answers
190 views
How can it happen to find infinite bases in $\mathbb R^n$ if $\mathbb R^n$ does not admit more than $n$ linearly independent vectors?
How can it happen to find infinite bases in $\mathbb R^n$ if $\mathbb R^n$ does not admit more than $n$ linearly independent vectors?
Also considered that each basis of $\mathbb R^n$ has the same ...
2
votes
2answers
66 views
How can we describe isomorphism in a tangible way?
What is the best way of describing isomorphism between two vector spaces? Is there a real life analogy of isomorphism?
2
votes
1answer
159 views
interpreting the power of adjacency matrix
Given a directed graph $G$, and let $A$ be $G$'s adjacency matrix, whose $(i,j)$-entry is 1 when there is an edge from $i$ to $j$.
Is there any interpretative meaning of the $(i,j)$-entry of the ...
2
votes
0answers
85 views
Intuition in permutations for Laplace Determinant Expansion
Starting with the Leibniz formula for the determinant, I wish to derive the Laplace (Cofactor) Expansion. At the risk of being overly verbose, please see the proof here. Now I understand the idea of ...
0
votes
0answers
136 views
Structure tensor of a function and the distribution of gradients
In computer vision, one often computes what's known as the structure tensor of an image. The structure tensor of a an image (i.e. a function) is a matrix that, I quote from Wikipedia.
"summarizes ...
11
votes
4answers
389 views
What makes elementary row operations “special”?
This is probably a stupid question, but what makes the three magical elementary row operations, as taught in elementary linear algebra courses, special? In other words, in what way are they "natural" ...
2
votes
1answer
116 views
Why should coordinate transformations be reversible?
Intuitively I understand why coordinate transformation should be reversible. New coordinates should cover the same area covered by the initial coordinates, i.e. there should be one-to-one mapping.
...
4
votes
2answers
112 views
The Duality Functor in Linear Algebra
I'm trying to gain an intuitive understanding of the following construction:
For any vector space $M$ over a field $R$, one can define the algebraic dual of $M$ as $M^* := \mathsf{Hom}(M, R)$ and ...
4
votes
2answers
293 views
what does following matrix says geometrically
Let $M\subset \mathbb C^2$ be a hypersurface defined by $F(z,w)=0$. Then for some point $p\in M$, I've
$$\text{ rank of }\left(
\begin{array}{ccc}
0 &\frac{\partial F}{\partial z} ...
3
votes
5answers
322 views
What's an intuitive way of looking at quotient spaces?
I understand the concept of $\mathbb{Z}/n\mathbb{Z}$, but I am having a really hard time understanding how this concept of quotients applies to vector spaces. Suppose $V = \mathbb{F}[x]$ is a vector ...
0
votes
2answers
633 views
what is the geometrical interpretation to positive definite matrix
What is the geometrical interpretation of positive definite matrix ?
(not necessarily symmetric)
if $A$ is positive definite, what does it do to a vector $x$ (i.e. $Ax$)?
3
votes
3answers
776 views
Bilinearity: what does it mean?
What does bilinear really mean? Everytime I heard the word, I think it should be "linear in 2 ways?"
For example, from the definition of inner product (taken from Appendix A of "Wavelets For ...
1
vote
1answer
207 views
Is there an intuitive interpretation of $ABA^T$?
I see expressions like this all the time in technical literature. The only $A$ and $B$ can be any size matrices as long as the expression is legal. I believe that the transposition is usually ...
5
votes
2answers
1k views
Examples for proof of geometric vs. algebraic multiplicity
Here you see a supposedly easy proof of a well-known theorem in linear algebra:
Although I know I should understand this, I don't :-(
Obviously there are too many indices and stuff, so I don't see ...
5
votes
7answers
3k views
practical uses of matrix multiplication
The use of matrix multiplication is usually given with graphics initially (scalings, translations, rotations, etc). Then there are more in-depth examples such as counting the number of walks between ...
4
votes
1answer
159 views
Visualizing Operators on $\mathbb{C}^n$
I am trying to get some better intuition about operators on complex inner product spaces. When we identify $\mathbb{C}^n$ with $\mathbb{R}^{2n}$, is there a nice geometric interpretation for the ...
4
votes
1answer
484 views
Explain what is a linear transformation
My textbook says "unique linear transformations can be defined by a few values, if the given domain vectors form a basis." However, that is all it says.
So can someone explain what a unique linear ...
11
votes
2answers
3k views
matrix multiplication: interpreting and understanding the multiplication process
I have just watched the first half of the 3rd lecture of Gilbert Strang on the open course ware with link:
http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/
It ...
5
votes
1answer
901 views
Intuitive explanation of the Fundamental Theorem of Linear Algebra
Can someone explain intuitively what the Fundamental Theorem of Linear Algebra states? and why specifically it is called the above? Specifically, what makes it 'Fundamental' in the broad scope of the ...
6
votes
1answer
859 views
intuition for complex eigenvalues
The eigenvalues of a rotation matrix are complex numbers. I understand that they cannot be real numbers because when you rotate something no direction stays the same.
My question
What is the ...
9
votes
6answers
2k views
How are eigenvectors/eigenvalues and differential equations connected?
In school and at university we never had eigenvalues nor differential equations, so these concepts were really giving me a hard time. Now I developed some intuition for both concepts.
I learned that ...
7
votes
2answers
618 views
The determinant is the integral of algebra. The integral is the determinant of analysis
This is probably an obvious parallel that most people are aware of, but I only just noticed it the other day and it made me quite excited. The determinant in algebra has a lot in common with the ...
2
votes
3answers
194 views
Intuition Behind Balanced Sets
Suppose $B \subset X$ where $X$ is a vector space. $B$ is called balanced if $\alpha B \subset B$ for every $\alpha \in \Phi$ with $|\alpha| \leq 1$. Note that $\Phi = \textbf{R}$ or $\Phi = ...
1
vote
3answers
258 views
Intuition behind tensor expansions of linear maps
Given finite-dimensional vector spaces $V,W$, there is an isomorphism $\text{Hom}(V,W) \rightarrow V^* \otimes W$. In particular, any linear map $\phi : V \rightarrow W$ has a tensor expansion $\sum ...
11
votes
3answers
1k views
The intuition behind generalized eigenvectors
An ordinary eigenvector can be viewed as a vector on which the operator acts by only stretching (without rotating) it.
Is there a similar intuition behind generalized eigenvectors?
EDIT: By ...
29
votes
4answers
2k views
Why do we care about dual spaces?
When I first took linear algebra, we never learned about dual spaces. Today in lecture we discussed them and I understand what they are and everything, but I don't really understand why we want to ...
10
votes
2answers
525 views
Is it misleading to think of rank-2 tensors as matrices?
Having picked up a rudimentary understanding of tensors from reading mechanics papers and Wikipedia, I tend to think of rank-2 tensors simply as square matrices (along with appropriate transformation ...
10
votes
4answers
3k views
Is there a geometrical interpretation to the notion of eigenvector and eigenvalues?
The wiki article on eigenvectors offers the following geometrical interpretation:
Each application of the matrix to an arbitrary vector yields a result which will have rotated towards the ...
8
votes
4answers
2k views
How to visualize a rank-2 tensor?
The notion (rank-2) "tensor" appears in many different parts of physics, e.g. stress tensor, moment of inertia tensor, etc.
I know mathematically a tensor can be represented by a 3x3 matrix. But I ...
131
votes
9answers
8k views
What's an intuitive way to think about the determinant?
In my linear algebra class, we just talked about determinants. So far I’ve been understanding the material okay, but now I’m very confused. I get that when the determinant is zero, the matrix doesn’t ...
