Articles of matrix calculus

Derivative of matrix exponential wrt to each element of Matrix

I have $x=\exp(At)$ where $A$ is a matrix. I would like to find derivative of $x$ with respect to each element of $A$. Could anyone help with this problem?

Convexity of a trace of matrices with respect to diagonal elements

Can we prove that $\mbox{trace}({\bf A} ({\bf P}+{\bf Q})^{-1} {\bf A}^T)$ is a jointly convex function of positive variables $[q_1,q_i,…,q_N]$, where ${\bf Q}=\mbox{diag}(q_1,…,q_N)$, $q_i>0 , \forall i$, and ${\bf P}$ is a positive definite matrix.

Differentiate vector norm by matrix

I’ve been trying to perform the following differentiation of a neural network: $$\frac{\delta||h(XW)\alpha-y||^2}{\delta W} = \frac{\delta}{\delta W}\sum_i(h(XW)_i\alpha-y_i)^2$$ Where $X$ and $W$ and matrices, $\alpha$ and $y$ are vectors, and $h$ is a point wise applied function. I’ve been reading the Wikipedia article on Matrix calculus and “The Matrix Cookbook” all day, but I can’t seem […]

Why the gradient of $\log{\det{X}}$ is $X^{-1}$, and where did trace tr() go??

I’m studying Boyd & Vandenberghe’s “Convex Optimization” and encountered a problem in page 642. According to the definition, the derivative $Df(x)$ has the form: $$f(x)+Df(x)(z-x)$$ And when $f$ is real-valued($i.e., f: R^n\to R$),the gradient is $$\nabla{f(x)}=Df(x)^{T}$$ See the original text below: But when discussing the gradient of function $f(X)=\log{\det{X}}$, author said “we can identify $X^{-1}$ […]

How can I derive euclidean distance matrix from gram-schmidt matrix?

This is my first post, sorry for my naiveness.. I know a basic equation that relates Gram-schmidt matrix and Euclidean distance matrix: $XX’=-0.5*(I-J/n)*D*(I-J/n)’$ Where $X$ is centered data (is $d \times n$), $I$ is identity matrix, $J$ is a matrix filled with ones (1), $n$ is the number of columns in $X$, and $D$ is […]

Possible eigenvalues of a matrix $AB$

Let matrices $A$, $B\in{M_2}(\mathbb{R})$, such that $A^2=B^2=I$, where $I$ is identity matrix. Why can be numbers $3+2\sqrt2$ and $3-2\sqrt2$ eigenvalues for the Matrix $AB$? Can be numbers $2,1/2$ the eigenvalues of matrix $AB$?

sum of all entries of the inverse matrix of a positive definite, symmetric matrix

Let $A$ be a positive definite, symmetric $n\times n$ matrix such that each entry $a_{i,j}$ of $A$ is a number in $[0,1]$ and $\text{diag}[A]=(1,1,\cdots,1)$. Does the sum of all entries of $A^{-1}$ have a finite bound? That is, $$ \text{sup}_{A}\sum_{1\leq i,j\leq n}\text{entry}_{i,j}[A^{-1}]<\infty? $$ What is the exact value of the supremum?

How prove this $|A||M|=A_{11}A_{nn}-A_{1n}A_{n1}$

This question already has an answer here: Determinant identity: $\det M \det N = \det M_{ii} \det M_{jj} – \det M_{ij}\det M_{ji}$ 1 answer

Is the matrix square root uniformly continuous?

Let $\operatorname{Psym}_n$ be the cone of symmetric positive-definite matrices of size $n \times n$. How to prove the positive square root function $\sqrt{\cdot}:\operatorname{Psym}_n \to \operatorname{Psym}_n$ is uniformly continuous? I am quite sure this is true, since on any compact ball this clearly holds, and far enough from the origin, I think the rate of change […]

Derivative of the trace of matrix product $(X^TX)^p$

Let $X$ be a squared matrix, We know that $\frac {\partial tr(X^TX)}{\partial X}$ is $2X$ But how about the case of $\frac {\partial tr((X^TX)^2)}{\partial X}$ or even $\frac {\partial tr((X^TX)^p)}{\partial X}$ Is there any generalization? Note that here $(X^TX)^2 = X^TXX^TX$ and similar case applies to $(X^TX)^p$