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Questions tagged [outer-product]

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4
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1answer
26 views

Write a bivector as the exterior product of two vectors

The Wikipedia article https://en.wikipedia.org/wiki/Bivector#Simple_bivectors states that "A bivector that can be written as the exterior product of two vectors is simple. In two and three ...
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0answers
22 views

In what spaces are outer product of x with itself positive semidefinite?

In the case of vectors in $\mathbb R^n$, it is quite simple to see that for any vector $x$, $$ v^Txx^Tv = (v^Tx)^2 \geq 0$$ so clearly the form $xx^T$ must form a positive semidefinite matrix. But ...
4
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1answer
33 views

Definition of outer product

I am trying to understand the concept of outer product in quantum mechanics. I read "Quantum Computing explained" of David MacMahon. I can understand the transition in (3.12): $$(|\psi\rangle \...
1
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1answer
54 views

Expected value of outer product of multivariate normal random vector with itself

Let's say I have a random vector $\boldsymbol{t}$ that is distributed according to a multivariate normal distribution: $$ \boldsymbol{t} \sim \mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Psi}) $$ I now ...
1
vote
1answer
150 views

Outer product reformulation of LU decomposition

For my numerical analysis class, I wanted to implement the rather simple, in-place LU decomposition algorithm. I did this the naive way and found my code was very, very slow, because I was doing every ...
0
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0answers
32 views

Decomposition of the Outer Product

I have an $m \times n$ matrix that is statistical data. I have a theoretical model about how that data is constructed in which the matrix is the outer product of two vectors. Let's call those vectors $...
2
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0answers
17 views

Find the solution of an outer product induced system

Sorry if the question is lame, but I'm struggling to find the answer to the following problem: Given a matrix $A\in \mathbb{R}^{n,n}$ and a column vector $b\in \mathbb{R}^{n}$, how can one find the ...
0
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0answers
18 views

Finding sums of minimal amounts of outer products?

What numerical methods exist for writing matrices as sums of as few self-outer products as possible? For example a matrix: $$\begin{bmatrix}1&2\\2&4\end{bmatrix} = \begin{bmatrix}1\\2\end{...
2
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2answers
24 views

Calculate matrix powering given one outer product: $(x\cdot{y}^T)^k$

It is an exercise on the chapter one of a book. Book: "Matrix Computations 4th edition" by Golub and Van Loan. It reads: Give an $O(n^2)$ algorithm for computing $C=(x\cdot{y}^T)^k$ where $x$ and $...
2
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1answer
2k views

Distinguishing between inner product and outer product in matrix notation

As a recent field transferee from chemist to data scientist, I find myself wading through more matrix multiplication than I'm used to. I did some linear algebra way back, but I struggle with ...
0
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0answers
32 views

Need help understanding the similarities and differences between various products.

What is the similarities and difference between the tensor product, outer product, exterior product, inner product, dot product, cross product? Trying to get into understanding all of them. Any ...
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0answers
46 views

Where did the “Outer” and “Inner” Product nomenclature come from?

I have recently become familiar with the concept of the outer and inner products on vector spaces - but why have they come to be called the "outer product" and "inner product"? What is the history ...
-1
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1answer
259 views

Why is the sum of outer products equal to the matrix product of a matrix and its transpose , so $A^TA = \sum_{i=1}^n a_i a_i^T$? [duplicate]

Why is the sum of outer products equal to the matrix product of a matrix and its transpose? So $A^TA = \sum_{i=1}^n a_i a_i^T$, where $A = [ a_0, a_1 , ... , a_n ] $, $a_i \in \mathbb{R}^k$. An answer ...
-1
votes
1answer
67 views

compute the inverse of matrix which is the Kronecker product of two vectors

I would like to compute the inverse of the following matrix \begin{equation} A=\begin{pmatrix} a^2b^2+\sigma^2&a^2bd &ab^2c&abcd\\ a^2bd &a^2d^2+\sigma^2&abcd&acd^2\\ ab^2c&...
0
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1answer
39 views

For a given $M$, how can we solve $M=w\vec 1 ^T + \vec 1v^T$?

Let a matrix $M\in\mathbb R^{m\times n}$ be given. Define $\vec 1 = [1, \ldots ,1]^T$ to be a vector of all ones (I don't specify its length since it will be clear from the context.) Under what ...
-1
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1answer
29 views

How do I use vectorization to simplify matrix integration problem?

Can someone show the detailed procedures for proof: \begin{equation*} \text{vec}\left(\int^T_0ds\,e^{-Ks}\Sigma\Sigma^\text{T}e^{-K^\text{T}s}\right) = \left(K\otimes I+I\otimes K\right)^{-1}\text{...
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0answers
25 views

Distribution of outer product of two uniform r.v.'s on ellipsoid

Suppose $Z \sim N(0, \Sigma)$ and define $\widetilde{Z} \overset{\text{def}}{=}Z/\|Z\|_2$. Is there a closed-form distribution for $P \overset{\text{def}}{=} \widetilde{Z} \widetilde{Z}^\intercal$? I'...
0
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1answer
175 views

Use of tensor product in calculation of vector projection

In a textbook I stumbled upon this statement: The projection $(\mathbf v \mathbf · \mathbf r)\mathbf r$ can be replaced by the tensor product $(\mathbf r ⊗ \mathbf r)\mathbf v$ Where $\...
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0answers
38 views

Question on the outer product used to construct this 2-Qbit state

I am reading a book on quantum computing. The author is constructing an arbitrary 2-Qbit state from unitary transformations. I need help understanding on step in his logic. For those unfamiliar with ...
2
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0answers
197 views

Outer product of cross product vector with itself

I'm wondering if there is another, possibly more efficient, way to get to the $3 \times 3 $ symmetric matrix $\mathbf{D}$ below from 3-vectors $\mathbf{a}$ and $\mathbf{b}$ then the straight forward ...
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2answers
1k views

Sum of the outer products of $n$ independent vectors is always full rank? [closed]

If I take $n$ linearly independent vectors ($n \times 1$), $v_1, v_2, \ldots, v_n$ and construct a matrix which is the sum of their outer products, $$M = \sum_i^n v_iv_i^t$$ Then, can it be proven ...
3
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0answers
295 views

Distribution of the outer product of two Gaussian vectors

Assume that $\mathbf{x} \sim \mathcal{N}(0,\mathbf{I}_n)$ and $\mathbf{y}\sim \mathcal{N}(0,\mathbf{I}_p)$ are two independent standard Gaussian vectors. What is the distribution of their outer ...
0
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1answer
41 views

Writing Matrix-Matrix products as sums

$X = USV^\top = \sum_{i=1}^{rank(X)}s_{ii}u_i(v_i)^\top$ I'm trying to wrap my head around this reformulation. $SV^\top$ does, of course, simply scale the rows of $V^\top$. So if we write $X=AB^\...
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0answers
128 views

Relating the eigenvalues of a sum of outerproducts after applying a change of basis

Let $A_1 \in \mathbb{R}^{k\times k}$ be a symmetric and positive definite matrix. Let $x_1,x_2 \in \mathbb{R}^k$. Suppose I have the outer product $x_1 x_1^T$, from this post we know that $x_1$ is an ...
0
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1answer
14 views

Does $-4xx^T+(\|x\|_2^2+1)I_n$ have Eigenvalue 0 for any $x\in\mathbb R^{n,1}$?

Is there any $x\in\mathbb R^{n,1}$ such that 0 is an eigenvalue of $A_x:=-4xx^T+(\|x\|_2^2+1)I_n$? If $A_x$ was of shape $\alpha xx^T+\beta I_n$ with $\alpha\geq0$ and $\beta>0$, this would be a ...
2
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1answer
1k views

Matrix $A^T A$ as sum of outer products

I have recently read in a script about statistical methods in a chapter about linear regression that for an $n \times k$-Matrix $A$ with $a_i$ as $i$. row of $A$: $$A^T A = \sum_{i=1}^{n} a_i^T a_i$$ ...
3
votes
3answers
538 views

How to prove this property of outer product?

I encountered the following equation about vector outer product: $$nn^T = [n]_{\times}^2 + I$$ where $n = [n_1, n_2, n_3]^T$ is a unit column vector, $I$ is the identity matrix, and $[n]_{\times}$ is ...
0
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1answer
303 views

Outer product of vector with self versus inner product

I have a real-valued column vector $v$ and the following equation $$ \frac{v v^T}{1+\alpha v^T v} $$ with $\alpha>0$, and I'm trying to find an upper limit not including $v$. I have found that the ...
5
votes
1answer
2k views

Is the outer product of a column vector with itself positive semi definite? [closed]

Say we have a column vector $x=[x_1\ x_2\ x_3]^T$. Then is $ xx^T $ positive semi definite.
0
votes
1answer
888 views

Derivative of outer product

I want to calculate the first generalized coordinate derivative $\frac{\partial}{\partial q}$ being $q=x$ or $q=y$ or $q=z$ of the outer product between two identical vectors $R=\bf{r}\bf{r}^{T}$ ...
0
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0answers
116 views

Outer product - Einstein notation - Is there a mistake?

In my book we defined $$(\mathbf{a}\otimes\mathbf{b})_{ij} = a_ib_j$$ then the book goes on and says the outer product is distributive. Then it does the following $$\mathbf{a}\otimes \mathbf{b} = (a_1\...