# 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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### Name for a matrix with sum of elements along every row/column is equal?

Is there a specific name for a symmetric matrix where sum of elements along every individual row is equal to the sum of elements along every individual column? For example, \begin{equation*} A = \...
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### Matrix Calculus: Derivative of double centered Euclidean distance matrix

I want to do this matrix calculus: Given, a distance matrix of squared Euclidean distances $D(X)_{n \times n}$ of $n$ points $X \in \mathbb{R}^k$, and given $C_n$, a centering matrix as defined in ...
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### Schur complement question.

Good evening. Before start, I would like to pray for the end of the COVID19 virus soon. I have a question related to the calculation of one Schur complement matrix. Here is the matrix I'm interested ...
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### Stuck on matrix derivative

I am stuck with this (probably simple) derivatives: $$\frac{\partial}{\partial X}Tr((A\odot(B^{T}XB))C)\;\;and \;\;\frac{\partial}{\partial X}Tr((A\odot(B^{T}XX^{T}B))C)$$ where $A,B,C$ are ...
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### Algorithms for computing $(I + \frac{Q}{n})^n$

I need to compute the matrix $(I + \frac{Q}{n})^n$ where $n = 2^k$, $k$ is integer. I want to know if there is any practical and effective algorithms.
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### Complex derivative of Hadamard product inside Frobenius norm

I'm trying to find the complex derivative of $$||R - P \circ \gamma \gamma ^H||_F ^2$$. with respect to $\gamma$. I saw the post regarding the real counterpart of the same question here. However, ...
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### Rearranging linear system containing a symmetric matrix

I'm trying to rearrange the following equation to get only Q on the LHS: $$A = -Q + t(Q M + M^T Q - kQ)$$ Q is a symmetric matrix. A is a known symmetric matrix, M is an unknown non-symmetric ...
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### Unique Solution for Square-root function on Matrices

Prove that for every n by n matrix M sufficiently close to the identity matrix there exists a square-root matrix (solution of $A^2 = M$) and the solution is unique if $A$ is required to be ...
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### A system of SDEs

The question: Suppose I have a $d$-dimensional stochastic process $X(t)$ such that $$dX = (\alpha + AX)dt + \sigma dW(t)$$ where $A$ and $\sigma$ are constant $d$-by-$d$ matrices, $\alpha$ is a $d$...
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### Taylor expansion of a function of a symmetric matrix

First of all let me tell you that the answer to this question is likely to confirm a not-so-minor error in a very popular (and excellent) textbook on optimization, as you'll see below. Background ...
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### Diagonalizable matrix over R

I had study about solving linear difference equation system by using results of linear algebra. But I have a problem with the following exercise: Suppose that $A$ is a matrix with real entries, $A$ ...
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### Derivative of L1 norm of Hadamard product

I am trying to find the derivative of $f(B)=\lambda\Vert W \bigodot B \Vert_1 + \frac{\rho}{2}\Vert A-B \Vert_F^2 + tr(\Delta^T(A-B))$ with respect to B. where B is (n×n）matrix, W is (n×n）constant ...
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### Least Squares: Derivation of Normal Equations with Chain Rule (Revisited)

My question pertains to someone else's answered question that has made me curious. The OP wanted to differentiate the following using the chain rule: $$J(\theta)=\frac12(X\theta-y)^T(X\theta - y)$$ ...
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### What does $[T]_{\cal BB}$ mean?

I have been given a matrix P where the columns represent a basis in B. I have also been given a matrix A which is the standard matrix for T. I am then supposed to calculate the matrix for T relative ...
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### How to write the first term of Taylor expansion for a function matrix?

Assume $F(A)=\left\| A^N v-w \right\|^2$ where $A$ is a square matrix and $v$ and $w$ are two constant vectors and $N$ is a positive integer. Also, the norm is the Euclidean norm. How to write the ...
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### Minimizing $\| x A - B\|_F^2$ With a Constraint

I have previously asked an optimization question Here. I will reiterate the question and simply add a constraint to it: I have 2 known grayscale images (256×256 matrices) $A$ and $B$ and want to find ...