Questions about understanding a problem geometrically.
Questions tagged [geometric-interpretation]
192 questions
227
votes
6 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
- 2,271
117
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
- 4,121
93
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…
Alf
- 2,667
45
votes
2 answers
What do zero eigenvalues mean?
What is the geometric meaning of a $3 \times 3$ matrix having all three eigenvalues as zero? I have interpretations in mind for $0$ , $1$ , and $2$ eigenvalues being zero, but what about all of them?
ThanksABundle
- 897
37
votes
3 answers
Geometric interpretation of the determinant of a complex matrix
A complex $n$ -dimensional vector space $V$ can be thought of as a real $2n$ -dimensional vector space equipped with a map $J:V \to V$ with $J^2 = -I$ . Complex-linear maps are then linear maps $V \to V$ which commute with $J$ . One can think of $J$…
Ash Malyshev
- 3,035
33
votes
4 answers
Geometric interpretation of $\frac {\partial^2} {\partial x \partial y} f(x,y)$
Is there any geometric interpretation for the following second partial derivative? $$f_{xy} = \frac {\partial^2 f} {\partial x \partial y}$$ In particular, I'm trying to understand the determinant from second partial derivative test for determining…
Lie Ryan
- 1,241
31
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 ||g||_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…
Asaf Shachar
- 26,337
26
votes
4 answers
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
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22
votes
2 answers
Geometric interpretations of matrix inverses
Let $A$ be an invertible $n \times n$ matrix. Suppose we interpret each row of $A$ as a point in $\mathbb{R}^n$; then these $n$ points define a unique hyperplane in $\mathbb{R}^n$ that passes through each point (this hyperplane does not intersect…
GMB
- 4,266
20
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
- 8,149
19
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 been…
theboombody
- 473
17
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 $i^2=-1$.…
David Zhang
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17
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
- 291
17
votes
5 answers
Geometric Interpretation of Determinant of Transpose
Below are two well-known statements regarding the determinant function: When $A$ is a square matrix, $\det(A)$ is the signed volume of the parallelepiped whose edges are columns of $A$. When $A$ is a square matrix, $\det(A) = \det(A^T)$. It appears…
Tunococ
- 10,483
16
votes
3 answers
Why does the leading coefficient of a quadratic trinomial resemble some sort of a slope?
It's known that a unique parabola of the form $y=ax^{2}+bx+c$ exists for any three distinct points, provided that the points are non-collinear and their $x$ coordinates are distinct. Consider the following three points: $A$ is a point with…
TheProver
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