# Questions tagged [geometric-interpretation]

Questions about understanding a problem geometrically.

106 questions
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### Corollary of Projection onto a closed convex set and geometric interpretation

I need help with geometric interpretation of this theorem and with the corollary of the theorem: Theorem: projection onto a closed convex set Let $K \subset H$ be a nonempty closet convex set. ...
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### simplex $B^{-1} \cdot A_j$ tableau interpretation

In an iteration of the simplex tableau implementation, what is the interpretation of the columns $B^{-1} \cdot A_j$ underneath each variable $x_j$?
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### Representation for circumferences intersection

I'm trying to write a formula that represents the intersections points for $n$ circumferences. All of these circumferences intersect them to each other. Is there a good representation to explicit this ...
3answers
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### Proof and Geometric intuition of $u \leq 2\ln(1+u)$ for $u \in [0,1]$.

How can I prove the inequality $$u \leq 2\ln(1+u)$$ for $u \in [0,1]$. What is the geometric intuition behind this inequality?
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### Geometric interpretation of the tensor product of two projectif spaces $\mathbb{R} P^n \otimes \mathbb{R} P^n$ [duplicate]

We define a Projective space of a vector space as follow : http://en.wikipedia.org/wiki/Tensor_product#Tensor_product_of_vector_spaces Given a vector space $V$ over a field $\mathbb{K}$, its ...
3answers
132 views

### Geometric or matrix intuition on $A(A + B)^{-1}B = B (A + B)^{-1} A$

I am curious about a seemingly simple identity in matrix algebra. Though matrix multiplication is not commutative (the classic example of noncommutativity, it does allow a commutativity of sorts ...
1answer
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### Multiplying a matrix with its eigenvectors stretches or contracts the vector without changing its “direction”. Is this true for complex eigenvalues?

I tried to prove this as follows - Suppose A is a square matrix with a complex eigenvalue $\lambda$ and its corresponding eigenvector x. Then, by definition Ax = $\lambda$x Angle between x and the ...
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### What is the geometric interpretation of $\displaystyle\lim _{s\to 1}\sin²(\zeta(s))+\cos²(\zeta(s))$?

it's seems that $\displaystyle\lim _{s\to 1}\sin²(\zeta(s))+\cos²(\zeta(s))$ dosn't exist as shown here by wolfram alpha , and because $\zeta(1)$ is undefined , My question is to seek if the titled ...
1answer
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### 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 ...
1answer
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### Geometric interpretation of a basic identity in complex analysis

Consider the identity $|z_1+z_2|^2+|z_1-z_2|^2=2(|z_1|^2+|z_2|^2)$. The proof follows from breaking up the L.H.S. using $|z|^2=z\bar{z}$, expanding the factors and cancelling put the common terms. I'...
1answer
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### Geometric interpretation and proof of a well-known algebraic statement

We have to show that : Given pairs of integers $\{a,b\}$ and $\{c,d\}$, there exists a pair of integer $\{u,v\}$ such that $(a^2+b^2)(c^2+d^2)=u^2+v^2$. Further, if $a,b,c,d$ are non-zero, and both of ...
1answer
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### Non-geometric interpretation of dot products and eigenvectors

The idea is to motivate the SVD for use in a recommender system. Consider a matrix $A\in \mathbb{R}^{f\times u}$ where $A_{ij}$ captures how user $j$ rates film $i$ (on a scale from 1-10, some ...
2answers
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### Hyperplane equation intuition / geometric interpretation

Hello I'm trying to get a geometric appreciation for this n-dimensional hyperplane equation : $\frac{1}{\left\lVert \hat w \right\rVert} \times\ (\hat w \dot\ \hat x) - d = 0$ where: x is a point ...
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### 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 ...