Questions tagged [matrix-calculus]

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

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Gradient of Frobenius inner product involving a homogeneous function

Let $f:\mathbb{R}^{m\times n} \rightarrow \mathbb{R}^{m \times n}$ be a differential function. I want to take the gradient of the Frobenius inner product between $X$ and $f(X)$ with respect to $X$. ...
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Closed form for summation of matrix multiplication by its transpose

Assume $A_{N\times{N}}$ is a square matrix. Is there a closed form for $$I+A+AA'+(AA')A'+((AA')A')A'+...$$ Essentially a closed form for multiplication of the matrix by its transpose for infinite ...
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Is there any formula for the inverse of a 3x3 matrix concerning its row vectors?

I have a 3x3 matrix $A$: $$ A = \begin{bmatrix} \vec a_0^T \\ \vec a_1^T \\ \vec a_2^T \end{bmatrix} \in \mathbb R^{3\times 3}, $$ where $\vec a_i \in \mathbb R^3$. Is there any formula of $A^{-1}$ ...
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How to compute the derivative of this matrix equation

The matrix $\mathbf{A}(c)$ with the dimension $M \times N$, $c$ is a scalar variable. The matrix $\mathbf{d}$ is a constant matrix with the dimension $M \times 1$. If the formula $\frac{d\mathbf{A}(c)}...
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Formula for $k$th order derivative of the determinant of a matrix?

Background Jacobi's formula tells us that $$\frac{d}{dt}\det A(t) = \operatorname{tr} \left( \operatorname{adj}(A(t)) \frac{dA(t)}{dt} \right) = (\det A(t)) \cdot \operatorname{tr} \left( A(t)^{-1} \...
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How to solve the least squares for A with both X and Y as matrices?

I’m trying to solve the following problem: I have a series of n-dimensional vectors x and another of m-dimensional vectors y, so that y = Ax, with A being a m,n ...
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Why does the matrix exponential $e^A$ always exist?

Why does $e^A$ always exist for any given $n \times n$ matrix $A$? I can't find anything discussing this question, which is quite suprising, since it is such a general question.
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Trying to understand why a certain matrix derivative is sparse

I'm having a hard time understanding matrice derivatives with respect to derivatives, and came upon the following exercise which I am not sure how to solve. Let there be matrices ${\bf X} \in \Bbb R^{...
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Confusion about convergence for the logarithm of a matrix

How can I prove that for a matrix $A$, $$ \text{log}A = \sum_{m=1}^\infty (-1)^{m+1} \frac{(A-I)^m}{m} $$ is absolutely convergent if $||A - I|| < 1$? (I'm using the Hilbert-Schmidt norm.) In Hall'...
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Matrix Inversion of $f(X) = (X^{-1}-A)$

We have a function $$ f(X) = X^{-1} - A, $$ and I have to show with the help of Newtons's method: $$ X_{k+1} = X_k - Df(X_k)^{-1}f(X_k) $$ that the iteration takes the following form for $f(X_k)$: $$ ...
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Derivative of the inverse of a symmetric matrix w.r.t itself

I'm trying take the derivative of a symmetrix matrix $\mathbf{C}$ with respect to itself. $$ \begin{equation} \frac{\partial \mathbf{C}^{-1}}{\partial \mathbf{C}} \end{equation} $$ Using the indicial ...
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Reverse engineering a matrix in RREF

I know that multiple matrices can have the same RREF, but a matrix has a unique RREF, I am trying to reverse engineer a 3x3 RREF matrix to get an abstract matrix of the same size that has 2 columns ...
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How to show two matrices are similar? [closed]

Show that the matrices A = ⎡ 1 1 1 ⎤ ⎢ 1 1 1 ⎥ ⎣ 1 1 1 ⎦ and B = ⎡ 3 0 0 ⎤ ⎢ 0 0 0 ⎥ ⎣ 0 0 0 ⎦ are similar? I know I should show there exist P- such that A=PBP-, but how do I find P, and is there a ...
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Derivative of diagonal matrix expression: $f(X)=\text{diag}(X)M^T\text{diag}^{-1}(MX)$

Let X be a vector $\mathbb{R}^{n\times1}$ and M be a constant matrix $\mathbb{R}^{n\times n}$, and given the function $f(X)$ how could i find the derivative of $f(X)$ with respect to $X$? In the ...
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Does $e^X=\lim_{n \to \infty}\left(\text{Id}+\frac{X}{n}\right)^n$ hold for matrices?

Let $X$ be a $d \times d$ real matrix, $d>1$. Is it true that $$ e^X=\lim_{n \to \infty}\left(\text{Id}+\frac{X}{n}\right)^n\,\,\,? $$ Edit: It seems that this question is a duplicate. To make it ...
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Preserving Hurwitz stability of block matrices with same eigenvalues of each block

Let us assume that $$A=-\text{diag}(d_1,\ldots,d_m) +Q \text{ is Hurwitz},$$ where $d_i>0, \forall i\in \{1,\ldots,m\}$ and $Q\in \mathbb R^{m \times m}$ is a possibly full matrix. Can we deduce ...
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how can I mathematically express the number of combination

I have set of values, $X = [x_1, x_2, ... , x_N]$, every values in $X$ is combined with another value in both signs, it means, $y = [ (x_1, x_1), (x_1, -x1), (x1,x_2), (x1,-x_2), ..., (x_1,x_N), (x_1,...
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Proving the equivalence of two notions of ODE for matrices

Let $\mathcal{H}$ be a complex finite-dimensional Hilbert space with basis $\mathcal{B} = \{e_{1},...,e_{n}\}$. Suppose that, for each $t \in \mathbb{R}$, $M(t)$ is a linear operator on $\mathcal{H}$. ...
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Differentiate the summation over an index of a rank 3 tensor.

$X^{i,j,\lambda}$ is a rank 3 tensor, $W^{i,j}$ is a matrix. Is it possible to analytically solve the following? $\frac{d}{dX}\left|| W - \sum_\lambda X \right||^{2}$ I know with matrices we can ...
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Find the derivative of matrix function

Consider the differentiable functions $g_{1}$, $g_{2}$,..., $g_{m}$ which are defined in $R^{n}$ and $$ G := \begin{bmatrix} g_{1} & g_{2} & ... & g_{m}\\ \end{bmatrix}^T $$ If $$ b(x) = ...
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Derivatives of Matrices in Lagrangian and LASSO Optimization with Shrinkage Operators

The goal is to write a program that solves for x in $$\textbf{min}\quad \|A\textbf{x}-\textbf{b}\|_1+\sigma \|\bf x \|_1$$ for $A\in \mathbb{R}^{m\times n}$. We assume that ${m}<{n}$ and ${\sigma}&...
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1 answer
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Column-wise partitioned matrix multiply by Kronecker Sum matrix

I saw one operation (stated below) in a proof and I don't think I completely understand its innate operation logic although I can guess the answer. The proof idea is: Suppose $\exists$ a non-singular ...
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Proof the limit of matrices means about $e^H$ (where matrix $H$ is self-adjoint)

This problem is from Chap.6 of Introduction to Matrix Analysis and Applications of Petz. Prove for self-adjoint matrices $H$, $K$ that $$ \lim _{r \rightarrow 0} \left(e^{r H} \#_{\alpha}e^{r K}\...
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1 answer
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What the rank of a matrix with the elements of one column to be infinity? [closed]

Suppose the $m\times m$ real matrix $A$ is positive definite, it is without doubt that $\mathrm{rank} (A) = m$. Now, if all the elements in the first column and first row are assigned to be infinity, ...
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Covariance matrix of Linear Transformation

I need help. I'm looking at matrix-vector multiplication. Both vector (x) and matrix (A) are random and independent. The vector has a mean and a covariance matrix. And the matrix has the mean and ...
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Calculate $\frac{\partial}{\partial \mathbf{A}}\lVert \mathbf{A}^{\top}\mathbf{AX}-\mathbf{X} \rVert _{F}^{2}$

Calculate $\frac{\partial}{\partial \mathbf{A}}\lVert \mathbf{A}^{\top}\mathbf{AX}-\mathbf{X} \rVert _{F}^{2}$ with $\mathbf{A}\in\mathbb{R}^{M\times N}$ and $\mathbf{X}\in\mathbb{R}^{N\times D}$, and ...
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Derivative of $F(X)G(X)$

Let $X \in \mathbb{R}^{n \times n}$,$F(X) \in \mathbb{R}^{n \times n}$,$G(X) \in \mathbb{R}^{n \times n}$.Does the following expression holds:$\frac{\partial F(X)G(X)}{\partial X}$=$\frac{\partial F(X)...
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Derivative of matrix-diag matrix product

I would like to take a derivative of the following expression wrt vector $x\in\mathbb{R}^d$ $$ W\mathrm{diag}(f(Ax+b)) $$ where $f$ is some smooth element-wise function, $A\in \mathbb{R}^{K\times d}$, ...
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Proving $\exp(A^{*}) = \exp(A)^{*}$

I have seen this post here on math stack discussing the proof that the exponential of the adjoint matrix is the adjoint of the exponential of the original matrix, that is $\exp(A^{*}) = \exp(A)^{*}$. ...
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Why matrix exponential in two different methods not matching?

Consider the following matrix: $$A=\left[\begin{array}{ll} 1 & 1 \\ 4 & 1 \end{array}\right]$$ We need to find $e^{At}$. Method $1.$ Th eigen values of $A$ are $3,-1$. I have Diagonalized the ...
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Minimizing 2-norm of a matrix

Suppose I want to minimize the following matrix norm: $$\min_{\alpha\in\mathbb{R}, \beta\in\mathbb{R}^{n\times 1}} ||A-\alpha B-c*\beta'||_2, $$ where $A, B \in \mathbb{R}^{m\times n}$ and $c \in \...
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2 answers
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Derivative of transpose of matrix

Let $X$ be a $n*n$ matrix with its entry indices increasing along every column.For example,when $n = 2$, $X =\left( \begin{matrix}x_1 & x_3\\ x_2 & x_4\end{matrix} \right)$.Let $\rm{vec}(X)$ ...
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How could I calculate the derivate an expression with a diagonal inverse function?

Given a vector $x=[x_1,x_2,...,x_n]$ and a matrix $Z$ with dimensions $n\times n$, the function $g(x)$ is described by:$\def\diag{\operatorname{diag}}$ $$ g(x)=\diag(x) \diag^{-1}(Zx)$$ Where $\diag(x)...
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summation as a matrix multiplication

Let $X$ be $n \times k$ matrix , $D$ is an Euclidean distance matrix $n \times n$ ($d_{ij} = \|x_i-x_j\|$) and $D' $ is just $n\times n$ matrix of realnumbers. Then i want to find a gradient for a ...
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Averaged matrix exponential inverse

Let $H$ be a given symmetric matrix, i.e., $H\in\mathbb{R}^{d\times d}_{\rm sym}$, $d\geq1$. The operator $T$ is given by $$TX=\int_0^1e^{(1-s)H}Xe^{sH}\mathrm{d}s,\qquad X\in\mathbb{R}^{d\times d}_{\...
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Computational complexity sparse outer product

Suppose that I have a matrix $A$ of dimension $m \times m$, which can be computed as: $$A= X D X^\top$$ where $X$ is a matrix of dimension $m \times n$, with $m > n$, and $D= \operatorname{diag}(d)$...
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Problem $2.9.6$ (Perko's ODE): Show $|Y(t)| \le |Y(0)|\exp\left(\int_0^t \|A(s)\|\, ds\right)$ for $Y' = AY$

Let $A(t)$ be a continuous real-valued square matrix of size $n\times n$. Show that every solution of the nonautonomous linear system (where $Y(t) \in \mathbb R^n$ for all $t$ in the domain of $Y$) $$...
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Integration of matrix form of Vasicek variance (Python/Matlab)

$X_t$ is a vector and follows the following Vasicek process. $$ dX_t=(\mu-K\cdot X_t)dt+\Sigma_x\cdot dZ_t \\ $$ what is the variance of $X_t$? In scalar form the answer is $\frac{\Sigma_x^2}{2\cdot K}...
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1 vote
2 answers
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Differentiating a matrix with respect to a vector

In multivariate linear model, I have come across the following matrix-valued function of $\beta \in \Bbb R^p$. $$\beta \mapsto(y-X\beta)(y-X\beta)^{T}$$ where matrix $X \in \Bbb R^{n \times p}$ and ...
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Power series for $(A+\epsilon I)^{-1}$ for large $\epsilon$

I am trying to figure out a power series for $(A+\epsilon I)^{-1}$ when $A$ is an invertible matrix and $\epsilon$ is large. The Neumann series can be used when $\epsilon$ is small. Is there something ...
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ODE for the inverse of fundamental solution

Let $A:[0,T]\to \mathbb{R}^{n\times n}$ be a continuous function, and let $\Phi$ satisfy $\frac{d}{d t} \Phi_t=A_t \Phi_t$ for all $t\in [0,T]$, and $\Phi_0=I_n$. I hope to prove that the solution to ...
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Is a unital completely positive map taking this form trace-increasing?

I just came across this problem, in which we consider a CP map $$ \Phi(\rho)=\sum_i K_i\rho K_i^\dagger. $$ Now, let us transform $\Phi$ into a unital map, by composing it with a Kraus rank-1 map $\...
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1 vote
2 answers
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Gradient of Quadratic Function

I'm currently considering a problem where we have data vectors $X_1,\dots,X_n\in\mathbb{R}^d$ with labels $y_i=\pm1$. I want to find the minimizer of $L(\theta)=\sum_i(1-y_iX_i^t\theta)^2$ by finding ...
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How to maximize the generalized Rayleigh Quotient when both numerator and denominator have summation

The objective function is defined as $$R(\mathbf{x}_i) = \frac{\sum_{i=1}^{M}\mathbf{x}_i^H\mathbf{A}_i\mathbf{x}_i}{\sum_{i=1}^{M}\mathbf{x}_i^H\mathbf{B}_i\mathbf{x}_i}$$ where $\mathbf{x}_i\in\...
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Matrix derivative of $|I-2iT\Sigma|^{-\frac{n}{2}}$

I just started learning matrix derivatives and I'm trying to find derivative of $$f(T) = |I-2iT\Sigma|^{-\frac{n}{2}},$$ where $T$ and $\Sigma$ are symmetric matrices. This is what I already have: $$\...
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Analatic functions with two variables applied to square matrices

I understand if square matrix $A=P_{1}^{-1}D_{1}P_{1}$, where $D_{1}$ is a diagonal matrix, we can write $f(A)$ as $P_{1}^{-1}f(D_{1})P_{1}$. My question is how to define $f(A,B)$ where $A=P_{1}^{-1}...
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Derivative of the product of a matrix scalar function and a matrix with respect to a matrix

How to take derivative of \begin{align*} \dfrac{\partial\left[\operatorname{tr}(\boldsymbol{A}^2)\cdot\boldsymbol{A}^3\right]}{\partial\boldsymbol{A}}=\,? \end{align*} with respect to a matrix $\...
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3 votes
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How to find the coefficients of the second eigenvector?

I have a $2\times 2$ real symmetric matrix: $$\begin{pmatrix} A & C \\ C & B \end{pmatrix} $$ and I know that the eigenvalues are: $$\lambda_{\pm} = \frac{1}{2}(A+B)\pm \frac{1}{2}\sqrt{(A-B)^...
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Calculating the unknown elements of a matrix based on a known matrix.

Suppose there is a 2$\times$2 matrix $M=\matrix[a\ ,b\ ;\ c\ ,d]$, which $$M \matrix[y_1\ ,\ y_2]^T=\matrix[y_1'\ ,\ y_2']^T$$ if the elements of the matrix M are known, is it possible to calculate ...
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How to calculate the derivative of a matrix with respect to a vector?

I want to find the derivative of $x^*M(x)x$ with respect to $x$. $x \in R^2$ is a vector and $M(x)$ is a smooth symmetric matrix function that maps $R^2 \to R^{2\times 2}$. I can write the matrix as $$...
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