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Questions tagged [matrix-calculus]

Matrix calculus is about doing calculus, especially derivative and infinite series over spaces of vectors and matrices.

Matrix calculus studies derivatives and differentials of scalar, vector and matrix with respect to vector and matrix. It has been widely applied into different areas such as machine learning, numerical analysis, economics etc.

There are basically two methods.

  • Direct: Regard vectors and matrices as scalar so as to compute in the usual way in calculus. And The Matrix Cookbook provides a lot of basic facts.

  • Component-wise: Write everything in indices notation and compute in the usual way componentwisely. Einstein summation convention is frequently used.

3828 questions
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What does the (∂X) symbol means in Matrix Cookbook?

Check page 8 and equation 34 of Matrix Cookbook. In this simple case we take the derivative of a matrix that is multiplied by a constant and we have: $\partial (aX) = a*(\partial X)$ Probably missing something crucial because I would expect ∂aX/∂X =…
George Pligoropoulos
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Compute $\frac{\partial}{\partial x} (x^T \exp(x x^T) x )$

How to compute $\frac{\partial}{\partial x} \big(x^T \exp{x x^T} x \big), ~x \in \mathbb{R^n}$? Any ideas
Clipper
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Matrix derivative formula using the matrix chain rule

Let $X \in \mathbb{C}^{m \times n}$ be a matrix. Let $F(X) \in \mathbb{C}^{m \times m}$ be a matrix, function of $X$, e.g. $F(X) = I_m + X X^{\dagger}$, where $^\dagger$ means conjugate-transpose and $I_m$ is the identity matrix of dimension $m$.…
PseudoRandom
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Proof: rank(A) ≤ k if A is the sum of k matrices of rank 1

Let A be an m × n matrix that is the sum of k matrices of rank 1. Prove that rank(A) ≤ k. I have learned this proof the other way around (given the fact that rank(A)=k, prove that A can be written as the sum of k matrices of rank 1). But I don't…
Emily
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Three proofs with row-equivalent matrices

I am struggling with these proofs, since I haven't done any proofs with row-equivalent matrices before, so I cannot get the idea behind it really. I was hoping you could help me out to get on the right track! Prove that two m x n matrices A and B…
Emily
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Tensor derivatives

Refer http://www.deeplearningbook.org/contents/mlp.html Page 207 Usually we do not apply the back-propagation algorithm merely to vectors, but rather to tensors of arbitrary dimensionality. Conceptually, this is exactly the same as…
Abhishek Bhatia
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Matrix equation solving

My question is very simple, but I it seems to be impossible for me to figure it out. Here we go, find $\mathbf Z$ such that $$ (\mathbf Z^t\mathbf Z)^{-1}\mathbf Z^t\mathbf W=X_1Y_1+X_2Y_2 $$ Where $\mathbf Z$, $\mathbf W$, $X_1$, $X_2$, $Y_1$…
Chamberlain Mbah
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matrix function differentiation

I have this simple function $Z=f(Q)=AQ$ where $A$ is a $1\times n$ matrix and $Q$ is $n\times m$ matrix of variables I want to calculate derivative of $Z$ with respect to $Q$. (In the reference that I read the answer is $A^T$. This equation is…
Anas Bari
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How to computer complex Matrix derivative

I want to solve following minimization problem using lagrange multipliers... i have problem computing derivatives w.r.t complex matrx K .. can anybody do it? $$\min_K \quad trace(KRK^{H}) \qquad \text{s.t. } KH=I $$ Thanks
alam
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Differential of a matrix function with Taylor development

I'm a beginner and I have a question. I'm sorry for my bad english. Let $A$ be an invertible n by n matrix, and let $F$ be a function defined on $M_n(C)$ by $F(X) = X^2 - A$. I would like to know how we can calculate $DF(X)(H)$ the differential of…
Mélanie De la Cheminée
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Square root of a matrix 3 without diagonalization

I have two questions. I want to know how I can say that this matrix admits an infinite number of square roots : $\begin{pmatrix} 0 & 0 & 1\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{pmatrix}$ And that this matrix doesn't admit square root. $\begin{pmatrix} 0 &…
Mélanie De la Cheminée
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What's the derivative of $X^TX$ w.r.t. $X$

What's the derivative of $X^TX$ w.r.t. $X$ Note that X is not a square matrix
Tony L
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Derivative of a Matrix Exponential with respect to A Matrix

I need to take the derivative of an exponential function with respect to a matrix. $\frac{\partial}{\partial A} (e^{At})$ where $t$ is a scalar and $A$ is a matrix not dependent on $t$. Yes those $A$'s are the same variable for clarification. Thank…
c smith
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Are the following matrix derivations correct? What are the involed rules?

Are the following two derivations correct? $$ \nabla_{\mathbf{w}}\mathbf{w}^T\mathbf{w} = 2 \mathbf{w} $$ $$ \nabla_{\mathbf{w}} ||\mathbf{y}-\mathbf{X}\mathbf{w}||_2^2 = 2\mathbf{X}^T(\mathbf{y}-\mathbf{X}\mathbf{w}) $$ (I assume that $\mathbf{y}$…
ndrizza
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Differentiating the residual sum of squares

Suppose that we have \begin{equation} (\boldsymbol y - \boldsymbol X \boldsymbol \beta)^T(\boldsymbol y - \boldsymbol X \boldsymbol \beta) \end{equation} where $\boldsymbol y$ and $\boldsymbol \beta$ are $n$ by $1$ column vectors and $\boldsymbol…
darkgbm
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