Questions about understanding a problem geometrically.
Questions tagged [geometric-interpretation]
158 questions
194
votes
4 answers
What is the geometric interpretation of the transpose?
I can follow the definition of the transpose algebraically, i.e. as a reflection of a matrix across its diagonal, or in terms of dual spaces, but I lack any sort of geometric understanding of the transpose, or even symmetric matrices.
For example,…
Zed
- 1,941
- 3
- 12
- 3
96
votes
7 answers
Geometric interpretation of $\det(A^T) = \det(A)$
$$\det(A^T) = \det(A)$$
Using the geometric definition of the determinant as the area spanned by the columns, could someone give a geometric interpretation of the property?
dfg
- 3,481
- 5
- 25
- 41
74
votes
6 answers
Geometric interpretation for complex eigenvectors of a 2×2 rotation matrix
The rotation matrix
$$\pmatrix{ \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta}$$
has complex eigenvalues $\{e^{\pm i\theta}\}$ corresponding to eigenvectors $\pmatrix{1 \\i}$ and $\pmatrix{1 \\ -i}$. The real eigenvector of a 3d rotation…
J..
- 2,337
- 1
- 22
- 26
24
votes
1 answer
Geometric interpretation of Hölder's inequality
Is there a geometric intuition for Hölder's inequality?
I am referring to $||fg||_1 \le ||f||_p ||f||_q $, when $\frac{1}{p}+\frac{1}{q}=1$.
For $p=q=2$ this is just the Cauchy-Schwarz inequality, for which I have geometric intution: The projection…
Asaf Shachar
- 23,159
- 5
- 19
- 106
19
votes
5 answers
Understanding Linear Algebra Geometrically - Reference Request
I know geometry and I know linear algebra but when I understand a linear algebraic concept geometrically, my head just explodes and things just become so much clearer and easier to understand...not to mention easier to remember or figure out its…
Fixed Point
- 7,524
- 3
- 28
- 46
18
votes
1 answer
Do the BAC-CAB identity for triple vector product have some intepretation?
As in the title, I was wondering if the formula: $$a\times (b\times c)=b(a\cdot c)-c(a \cdot b)$$ for $\mathbb R ^3$ cross product has some geometrical interpretation. I've recently seen a proof (from Vector Analysis - J.W. Gibbs) that's not at all…
pppqqq
- 2,030
- 17
- 23
17
votes
5 answers
Geometric Interpretation of the Basel Problem?
Does the Pi in the solution to the Basel problem have any geometric significance? Every time I see Pi, I have to think of a circle. I would love to see a nice intuitive picture connecting the Basel problem with geometrical figures. Anyone here…
theboombody
- 301
- 2
- 5
13
votes
1 answer
How to understand the exponential operator geometrically?
Consider the geometric interpretation of an orthogonal matrix, a projection matrix, a (Householder) reflector, or even just matrix-vector multiply in general.
A matrix takes a vector from a vector space (after a basis has been fixed) and performs a…
Fixed Point
- 7,524
- 3
- 28
- 46
12
votes
10 answers
Why is it that $\int_a^b \int_c^d f(x)g(y)\,dy\,dx=\int_a^b f(x)\,dx \int_c^d g(y)\,dy$?
The title sums it up. It's simple to prove, but I'm wondering if there is a geometric interpretation?
user142299
12
votes
3 answers
Geometry of the dual numbers
A dual number is a number of the form $a+b\varepsilon$, where $a,b \in \mathbb{R}$ and $\varepsilon$ is a nonreal number with the property $\varepsilon^2=0$. Dual numbers are in some ways similar to the complex numbers $a+bi$, where…
David Zhang
- 8,302
- 2
- 35
- 56
12
votes
2 answers
Truly intuitive geometric interpretation for the transpose of a square matrix
I'm looking for an easily understandable interpretation for a transpose of a square matrix A. An intuitive visual demonstration, how $A^{T}$ relates to A. I want to be able to instantly visualize in my mind what I'm doing to the space when…
evolution
- 251
- 2
- 7
10
votes
2 answers
Geometric Interpretation of Rearrangement Inequality
We know that many of the famous classical inequalities have geometric interpretations. Can you give a geometric interpretation of Rearrangement Inequality?
Note: Rearrangement Inequality is
$$x_ny_1 + \dots + x_1y_n \leq x_{\sigma(1)}y_1+ \dots +…
scarface
- 1,443
- 11
- 24
10
votes
7 answers
Why does multiplication act like scaling and rotation of a vector in the complex plane?
I regularly use the geometric analogy of multiplication by a complex number to represent a scaling and rotation of a vector in the complex plane. For a very simple example, i would point up along the Y axis and multiplying it by i again would be a…
William Grobman
- 293
- 2
- 8
10
votes
0 answers
Gauss-Bonnet theorem proof considering membrane Force and hydrostatic fluid Pressure equilibrium
Is it possible to prove Gauss-Bonnet Theorem by using physics (Mechanics of materials) models?
For example in mechanics could one consider static equilibrium by action of hydrostatic pressure normal to a patch, tension in the curved patch…
Narasimham
- 36,354
- 7
- 34
- 88
9
votes
2 answers
Geometrical interpretation for the sum of factorial numbers
I am in need of a way to represent the sum
$1! + 2! + 3! + 4! = 1 + 2 + 6 + 24 = 33$
in a geometrical way. What I mean by this is that for example, the sum
$1^2 + 2^2 + 3^2 + 4^2 = 1 + 4 + 9 + 16 = 30$
can be represented geometrically as a pyramid…
Sigfrid Stjärnholm
- 103
- 4