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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why this equality involving matrix holds true?

I am studying Lemma 11 of the paper I am having difficulty understanding on the last step, in particular, I have two questions: The first question (1) On the first equality of the last step on page 15,...
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Can anyone help with this matrix problem? [duplicate]

enter image description here In this image, there's a question about asking to get the determinant of matrix D. I'm stuck on how to represent it as using only a and n.
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Derivative with respect to a rectangular matrix

Consider the following; $$J = L^{\text{T}}KR$$ where $$L \in \mathbb{R}^{p}, ~~ K \in \mathbb{R}^{p \times m}, ~~ r \in \mathbb{R}^{m}$$ I am attempting to find $\frac{dJ}{dK}$. This is what I ...
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What is the definition for the dot product of a co-vector and a dyadic product of vectors?

The definition of the dot product of a second order tensor with a vector is: $$(\mathbf{a} \otimes \mathbf{b}) \cdotp \mathbf{x} = (\mathbf{b} \cdotp \mathbf{x})\mathbf{a}$$ The definition of the ...
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Derivative of $\mathrm{Tr}({\boldsymbol \Omega}\boldsymbol H(\mathrm{diag}(\boldsymbol u))^H\boldsymbol H(\mathrm{diag}(\boldsymbol u)))$

Noting $H\in\mathbb{C}^{M\times M}$ with the following form of: \boldsymbol H=(\boldsymbol h+\boldsymbol g\mathrm{diag}(\boldsymbol u)\boldsymbol G)^{T}(\boldsymbol h+\boldsymbol g\...
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Intuition for vector calculus

In my statistics class, I was introduced to Fisher Information. As it comes from the Taylor Expansion in vector form, I wanted to know terms were ordered in a certain way - whether it was just to make ...
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Derivative of multiplied vectors and matrix with respect to single variable in vector

Let v be a vector $v = (a + bc_1, ..., a + bc_n)$. Thus, we can say $v = a1 + bc$, where $1 = (1,..., 1)$ and $c = (c_1,...c_n)$. Let $M$ be a symmetric and quadratic matrix of proportions $n * n$, ...
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Partial Derivative of Matrix Summation with respect to a component.

I am trying to take the partial derivative of the following expression with respect to the component $[A]_{i, j}$, where both matrices $A$ and $B$ are $d\times d$, and $B$ is a diagonal matrix (this ...
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On the Hessian of the Log-Determinant and the solution provided in Stephen Boyd's textbook

This is a follow up to another question on the second-order approximation to log-determinant in Boyd's textbook, the excerpt can be found here: Here, $f$ is the log-determinant, $f(Z) = \log(\det(Z))$...
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Consider the following function: $$f(T) = \| T^{T}TB - C\|^2_2$$ where $T, B,$ and $C$ are all complex matrices. Let $T = X + iY.$ I wish to compute $\nabla f$ i.e. $\dfrac{\partial f}{\partial T}$....