[Math] Outer product of a vector with itself

linear algebra

Is there a special name for an outer product of a vector with itself? Is it a special case of a Gramian? I've seen them a thousand times, but I have no idea if such product has a name.

Update:
The case of outer product I'm talking about is $\vec{u}\vec{u}^T$ where $\vec{u}$ is a column vector.

Does is have a name in the form of something of $\vec{u}$?

Cheers!

Best Answer

In statistics, we call it the "sample autocorrelation matrix", which is like an estimation of autocorrelation matrix based on observed samples.

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Vector Cross Product – Defined Only for 3D?

Yes, you are correct. You can generalize the cross product to $n$ dimensions by saying it is an operation which takes in $n-1$ vectors and produces a vector that is perpendicular to each one. This can be easily defined using the exterior algebra and Hodge star operator http://en.wikipedia.org/wiki/Hodge_dual: the cross product of $v_1,\ldots,v_{n-1}$ is then just $*(v_1 \wedge v_2 \cdots \wedge v_{n-1}$).

Then the magnitude of the cross product of n-1 vectors is the volume of the higher-dimensional parallelogram that they determine. Specifying the magnitude and being orthogonal to each of the vectors narrows the possiblity to two choices-- an orientation picks out one of these.

[Math] Vector outer product for more than two vectors

In Python, NumPy provides a very useful function that typically solves this problem in one line: numpy.einsum (for Einstein summation convention). Quoting the doc:

Using the Einstein summation convention, many common multi-dimensional array operations can be represented in a simple fashion. This function provides a way compute such summations. The best way to understand this function is to try the examples below, which show how many common NumPy functions can be implemented as calls to einsum.

Tricky detail: The alphabetical order matters: np.einsum('am',A)outputs $A$, while np.einsum('ma',A) outputs $A^\top$.

In your case:

import numpy as np 

d = 10 
U = np.random.rand(d) 
V = np.random.rand(d) 
T = np.random.rand(d) 
R = np.einsum('ai,aj,ak->ijk',U,V,W)

Best Answer

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Vector Cross Product – Defined Only for 3D?

[Math] Vector outer product for more than two vectors

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