Questions tagged [outer-product]

Given two vectors $\vec{u},\vec{v}$ of dimensions $m,n$ respectively, their outer product $\vec{u}\otimes \vec{v}$ is the $m\times n$ matrix $M$ with entries $m_{ij}=u_i v_j$.

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Relation between pairwise outer products and Kronecker tensor product

Let $A,B$ be real $n\times n$ matrices (if we want complex entries, replace transpose with hermitian conjugate). Write them as an array of columns so that $A=\begin{pmatrix} A_1~|&\cdots& |~...
TheEmptyFunction's user avatar
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Outer product of basis vectors

Can someone please help clarify how to calculate the outer product of two basis vectors, assuming $\mathbb{R}3$? From the definition, I understand that $\textbf{a} \otimes \textbf{b}$ $= a_ib_j = c_{...
Alexander Savadelis's user avatar
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If a matrix is an outer product of two vectors; can I determine the vectors? [closed]

I am working with floating point numbers. There is a 3x3 matrix that has determinant 1e-14. I have reason to believe this matrix is an outer product of two vectors. If the assumption is correct, how ...
Mikke Mus's user avatar
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Outer product of row vectors. Does $\mathbf{x}^T \otimes \mathbf{y}^T$ = $\mathbf{x} \otimes \mathbf{y}$?

Does $\mathbf{x}^T \otimes \mathbf{y}^T$ = $\mathbf{x} \otimes \mathbf{y}$?
Tomek Dobrzynski's user avatar
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Why can we substitute $ V_{\mu \nu} $ to $ V_{\mu ; \nu} $ while inducing contracted Bianchi identity?

After $ ( A_\mu B_\nu )_{; \sigma ; \rho} - ( A_\mu B_\nu )_{; \rho ; \sigma} = A_\alpha B_\nu R^\alpha_{\mu \rho \sigma} + A_\mu B_\alpha R^\alpha_{\nu \rho \sigma} $ where $ A_\mu B_\nu $ is outer ...
posfn0319's user avatar
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Is $(\pmb{b}^\mathrm{T}\pmb{x})^2=\pmb{b}^\mathrm{T}(\pmb{x}\pmb{x}^T)\pmb{b}$ true?

Let $\pmb{b}$ and $\pmb{x}$ be two vectors in $\mathbb{R}^n$, then: $$(\pmb{b}^\mathrm{T}\pmb{x})^2=(\pmb{b}^\mathrm{T}\pmb{x})(\pmb{b}^\mathrm{T}\pmb{x})=(\pmb{b}^\mathrm{T}\pmb{x})(\pmb{x}^\mathrm{T}...
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Derivation of the contravariant basis vectors for the tangential place of a surface

We know that (see for example "Vector and Geometric Calculus" by Alan Macdonald, pp. 74, problem 5.4.2) given the covariant basis vectors $\mathbf{b}_1$ and $\mathbf{b}_2$ of the tangential ...
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Geometric Algebra: show $ A_r \cdot B_s = (-1)^{r(s-1)} B_s \cdot A_r$ and $A_r\wedge B_s = (-1)^{rs} B_s \wedge A_r$ from Hestenes and Sobczyk's book

I'm making a go at self-study from Hestenes and Sobczyk's book Clifford Algebra to Geometric Calculus. I'm stuck on the simple formulas in the first section for reversing the order for the inner and ...
Kyle Taljan's user avatar
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Counterexamples when outer products of matrices are the same other

Let X and Y be two different $m\times n$ matrices, what are the cases/examples when $XX^T= YY^T$ other than $X=YQ$, where Q is an orthogonal matrix? In other words, is there an example such that $X\...
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For what types of objects is the outer product defined in geometric algebra?

I just started to learn geometric algebra from the "Geometric Algebra for Physicists" book. Authors first give definition of outer product of two vectors $a \wedge b$. Then they give ...
Viacheslav Radko's user avatar
2 votes
1 answer
66 views

Reconstruct a vector from outer product with itself [closed]

Suppose I am given the matrix $A = xx^\top$ where $x \in \mathbb{R}^n$ is some vector that is unknown. How can I reconstruct the vector $x$ from $A$?
Azgen's user avatar
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Operations and properties of the 'additive' equivalent of the outer product of two vectors

I have two vectors 'A1' and 'A2' of size M and N. I have an algorithmic problem related to the 'additive outer product' (rather than the usually 'multiplicative') matrix of the two vectors $$A'(i,j) =...
persiflage's user avatar
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Likelihood maximization between an outer product of a probability vector and a deterministic binary matrix

We have a deterministic symmetric binary matrix (e.g., adjacency matrix) $A \in \{0, 1\}^{n \times n}$. Given a vector $a \in [0, 1]^{n}$, we compute its outer product $\tilde{A} = a a^T \in [0, 1]^{n ...
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What is $\frac{d}{d\mathbf v} \left[ \nabla \cdot (\mathbf v\otimes \mathbf v ) \right]$?

Let $\mathbf v(\mathbf x): \mathbb R^3 \to \mathbb R^3$ with $\mathbf x \in \mathbb R^3$. I am trying to determine the following gradient: $$ \frac{d}{d\mathbf v} \left[ \nabla \cdot (\mathbf v\otimes ...
berrygreen's user avatar
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given the dot and cross products of vectors b and v, solve for v

Given that the dot product of b and v is k, and that the cross product of b and v is c, solve for the vector v in terms of b, c, and k, of which are known and fixed. It may be useful to note that the ...
Joseph_Kopp's user avatar
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Why can these two expressions be equal in the curl and divergence?

I have the same question in here: https://physics.stackexchange.com/questions/741463/why-can-these-two-expression-be-equal-in-the-curl-and-divergence But honestly i am not sure that this question ...
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Bound the difference of two outer products $\| u u^T - v v^T \|_F$ [duplicate]

Given two row-vectors $u,v\in[-1,1]^n$, I would like to upper bound the Frobenius norm of the difference of their outer products. Formally, I would like an upper bound on $ \| uu^T - vv^T \|_F$. The ...
tranisstor's user avatar
2 votes
2 answers
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Why is $a \times b$ the null vector of the anti-symmetric matrix $M = ab^{\mathrm T} - ba^{\mathrm T}$?

Let $a$ and $b$ be two vectors of size $3 \times 1$. The matrix $M = ab^{\mathrm T} - ba^{\mathrm T}$ is $3 \times 3$ anti-symmetric (easy to prove), and also $a \times b$ is its null vector. I have ...
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How to represent the symmetric part of the outer product of two vectors represented as the outer product of two vectors?

Consider a matrix, $$M = x y^{\top}.$$ Say, $$M_{\text{sym.}} = xy^{\top}+yx^{\top}$$ is the symmetric part of $M$. How can one represent $M_{\text{sym.}}$ in the form $$M_{\text{sym.}} = zz^{\top},$$ ...
Abhiram V P's user avatar
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Where am i wrong in these two proofs? $\vec A \times (\vec B \times \vec C)=-(\vec A \cdot \vec C)\vec B +(\vec A \cdot \vec B)\vec C $

I want to prove $\vec A \times (\vec B \times \vec C)=-(\vec A \cdot \vec C)\vec B +(\vec A \cdot \vec B)\vec C $,here are my two proofs below $\vec A \times (\vec B \times \vec C)=m\vec B+n\vec C$ $\...
user16266657's user avatar
4 votes
2 answers
429 views

Why does the square of a bivector give its magnitude squared?

I'm trying to learn geometric algebra, but I've gotten stuck on the magnitude of a bivector. In this video, the following derivation is made: For $u,v \in \mathbb{R}^n, u^2v^2 = uv^2u = uvvu$ $= (u\...
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What type of product? $\mathbf{a} \circledast \mathbf{b}:= \mathbf{a}\mathbf{b}^T - \mathbf{b}\mathbf{a}^T$

What do you know about the names and properties of the products of the following definitions? \begin{align} &\mathbf{a,b}\in\mathbb{R}^N, \mathbf{C}\in\mathbb{R}^{N\times N}\\ &\mathbf{C} = \...
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Inverse of the outer product of some vectors with their transpose

Assume i have $n$ 3D unit vectors $v_s$, with different values. Then i define a matrix $T$ as: $$ T = \frac{1}{n} \sum_{s=1}^{n} v_s \times v_s' $$ where $v_s$ are $3\times 1$ vectors and therefore $...
Denis's user avatar
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8 votes
1 answer
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Intuitive understanding of oriented volume and trivectors

I get that the way a vector's arrowhead points corresponds to its orientation for a given direction (line). We can also understand vectors within $V$ as isomorphic to a set of endomorphic translations....
Tristan Duquesne's user avatar
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1 answer
63 views

Solving matrix equation $x^{T}Ay$ for $A$ exponential

Is there a standard way to approach solving bilinear expressions like below, where the parameter of interest, the only unknown, forms part of the matrix of the bilinear form $$x^{T}Ay=\beta$$ where $A$...
rbarc's user avatar
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3 votes
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eigenvalues of outer product matrix.

Let $\mathbb R^m\ni \mathbf x \ne \mathbf 0 \ne \mathbf y\in\mathbb R^n$. Let $A = \mathbf x\mathbf y^T$ and find the single non-zero eigenvalues of $A$. Note the compact SVD of this matrix $A$ is $$\...
user10101's user avatar
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Geometric proof of the Cross Product magnitude (without using sine and additional assumptions)

Given 3D vectors $\mathbf a=(a_1,a_2,a_3),\mathbf b=(b_1,b_2,b_3)$, the cross product is defined as: $$ \mathbf a\times\mathbf b = \begin{vmatrix} \mathbf i&\mathbf j&\mathbf k\\ a_1&a_2&...
J3soon's user avatar
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1 answer
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Show that $(I-\mathbf x\mathbf y^T)^{-1} = I-\frac{1}{\mathbf x^T\mathbf y- 1}\mathbf x\mathbf y^T$

Let $x,y\in\mathbb R^n$ and suppose that $x^Ty \neq 1$. Show that $(I-\mathbf x\mathbf y^T)^{-1} = I-\frac{1}{\mathbf x^T\mathbf y- 1}\mathbf x\mathbf y^T$. Note, I need to compute this directly not ...
user10101's user avatar
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Prove that $\{v_kv^\dagger_l\}$ are linearly independent if $A=\sum_k \lambda_kv_k v^\dagger_k$

The spectral decomposition of a positive definite matrix $A=\sum_k \lambda_kv_k v^\dagger_k$ where $\lambda_k>0$ and $v_k$ are orthonormal vectors. How do we prove that the set $\{v_kv^\dagger_l\}$ ...
Sooraj S's user avatar
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2 votes
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Outer Product Between Two Vectors of Arbitrary Dimensions

I am currently reading the first chapter of Geometric Algebra for Physicists and while I am quite familiar with abstract definitions of inner products and have quite a bit of abstract linear algebra ...
Chris's user avatar
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How Am I Wrong? Geometric Algebra Question

I'm currently teaching myself Geometric Algebra out of the textbook Geometric Algebra for Physicists, and there's an example I get wrong. In the example, they show that $(\mathbf{a}\wedge\mathbf{b})^2=...
Eccentric Tuber's user avatar
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691 views

Derivative of outer product of two vectors

Given two vectors $u(x)$ and $v(x)$ for $x\in\mathbb{R}^n$ and $u(x), v(x)\in\mathbb{R}^m$ what is the derivative of their outer product? $$ \frac{d}{dx} u(x) v(x)^\top = ? $$ Perhaps it is this? $$ \...
Euler_Salter's user avatar
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16 votes
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Intuitive explanation of outer product

The inner product between two vectors is the product of length of first vector and the length of projection of second vector on to the first vector. When I take an outer product its result is a matrix....
Leo's user avatar
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Efficient computation of determinant of identity minus outer product [duplicate]

Suppose $\hat{x}\in\mathbb{R}^d$ is a normalized vector and $\alpha\in(0, 1)$ a scalar. Is there a way to compute the determinant of the following matrix quickly? $$ I_d - \alpha\hat{x}\hat{x}^\top $$ ...
Euler_Salter's user avatar
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1 vote
1 answer
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How do you express the product of three matrices as a weighted sum of outer products? [closed]

How do you express the matrix ABC as the weighted sum of outer products of vectors extracted from A and C and with weights coming from matrix B? I'm not sure how to approach this problem. I tried the ...
Tolga Yilmaz's user avatar
1 vote
3 answers
2k views

Use of the quotient law for tensors by contraction

I'm having a hard time seeing how the quotient law for tensors is used it the following example of Riley, Hobson and Bence (3rd). Preceding the example, it reads Use of the quotient law to test ...
Mussé Redi's user avatar
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Condensed SVD decomposition of an outer product

Let $A = uv^{T} \in \mathbb{R}^{m \times n}$. Find the (condensed) SVD decomposition of $A$. Theorem (Condensed SVD decomposition) Let $A \in \mathbb{R}^{n \times m}$ be a non-zero matrix of rank $r$....
hexaquark's user avatar
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Simplifying an algorithm with matrix algebra. [closed]

Sorry in advances for my lack of knowledge on this topic. Given a $p$ by $p$ positive semidefinite matrix $S$ from which, I want to compute this algorithm ...
POC's user avatar
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Decomposing an inner product of matrices into outer products

In a lecture on bayesian reasoning and active learning I came across a slide that seemed to imply that the inner product of a matrix X (of n rows and d columns) with itself could be decomposed into ...
user3163829's user avatar
2 votes
1 answer
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Difference between dot product, inner product, cross product, outer product. And what are their symbols?

When I was learning machine learning, I often encounter these terms, yet I don't the difference and relationship between them. And what are the symbols for them? Can someone help me?
Yunchao 'Lance' Liu's user avatar
1 vote
1 answer
4k views

Proof that $x x^T$ is symmetric and positive semidefinite

For a generic vector $x\in\mathbb{R}^n$ it is clear the outer product $xx^T$ is a symmetric matrix. But how can we prove that this is a positive semidefinite as well? The matrix $X:=xx^T$ is given by ...
swissy's user avatar
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Can a tensor's indices be flipped arbitrarily?

I made use of the following intermediate step in a demonstration I did: $$...=a_i\sigma_{kj}b_k=a_i\sigma^T_{jk}b_k=...$$ where the "T" just shows that the indices have been flipped. After ...
Ian's user avatar
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2 votes
1 answer
159 views

Do Scalar Product, Dot Product, and Cross Product operands have special names?

Our basic operations have names for their operands: Addition: $\rm{Augend}+\rm{Addend}=\rm{Sum}$ -- Generally, we call them both $\rm{Addends}$ or $\rm{Summands}$. Subtraction: $\rm{Minuend}-\rm{...
Oliver's user avatar
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Is there a counterpart to the outer product? I.e., divide every term of a vector by every other term of another vector.

Regarding the dyadic product (specifically, the outer product or tensor product), notated as $\otimes$, can you provide comment on its counterpart? I'm not sure if it has a name, or if it has matrix ...
Armadillo's user avatar
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1 answer
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What is the intuition behind the outer product of two eigenvectors?

I know that the outer product of every two eigenvector forms a 2-D basis for the 2-D matrices. For example, when we write a matrix based on its eigenvectos, we have: $$ X = \sum_{i,j} \lambda_{i,j}...
user137927's user avatar
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Equality of inner product of outerproducts of vector with itself d-times and the d-th power of the inner product of the two vectors

Suppose $\mathbf{a}, \mathbf{b} \in \mathbb{R}^{n}$, proof that $\left\langle\mathbf{a}^{\otimes d}, \mathbf{b}^{\otimes d}\right\rangle=\langle\mathbf{a}, \mathbf{b}\rangle^{d}$ where $\mathbf{a}^{\...
Freddy Mixon's user avatar
2 votes
0 answers
274 views

Showing Gramian Matrix Positive Definite Using Outer Product

I want to show that the Gramian Matrix is positive definite. My idea is that this matrix can be written as an outer product of two vectors. So, let $(\cdot \ , \cdot)$ be defined as the Euclidean ...
ABCCHEM's user avatar
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107 views

How to solve matrix equation using outer product?

I have a matrix equation $A \cdot B \cdot C = D$ with $A$ being $1\times n$ row vector, $B$ a $n\times n$ matrix and $C$ a $n\times1$ column vector. $D$ is a number. I want to calculate $n\times n$ ...
Sayandip Ghosh's user avatar
1 vote
1 answer
288 views

Tensor spaces / outer product

Most of the examples I have ever seen use Cartesian vectors and matrix representation to illustrate this idea. That's fine and good, and easy to follow, but not very helpful for more abstract cases. ...
user3372039's user avatar
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1 answer
152 views

Bilinear Maps between Finite-Dimensional Vector Spaces & the Outer Product

Theorem. A function $f : \mathbb{R}^{n_1} \times \mathbb{R}^{n_2} \to \mathbb{R}^{n_3}$ is bilinear iff each component of $f(v,w)$ is a linear combination of terms of the form $v_iw_j$, where $v=\...
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