# Tagged Questions

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### Derivative with respect to a function

We have a function ${f(s,{\psi(s)}_{3\times 1})}_{3\times1}\tag1$ Given Data $f,\psi$ are matrices and their dimensions are already given in the question s is not a matrix, it is a scalar ...
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### Function with constant derivative

We have a column matrix $P_i$ defined as follows $P_i= {\begin{pmatrix} a_i \\ b_i \\ c_i \end{pmatrix}}_{3\times 1}\tag 1$. Given Data All $a_i,b_i,c_i$ are constants It is given that $i$ can ...
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### How can I calculate this matrix differentiation?

I am studying about the Matrix Differentiation. I don't know if this red box differential metric, which is how it is calculated.
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### Matrix Algebra - Linear dependency

We have a given equation $\frac{\mathrm{d}R(t) }{\mathrm{d} t}=R(t) \{(1-t)U_0+t U_1\}\tag 1$, all variables except scalar variable 't' has dimension $3 \times 3$. Given data $R(t)$ is ...
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### Definite Integral involving matrices

We have a definite integral of the form given below $f(t) = \int_0^1 e^{\alpha X(t)} \frac{dX(t)}{dt} e^{(1-\alpha) X(t)}\,d\alpha \tag 1$ Given Data in the question $X(t)$ is a ...
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### Derivative of determinant of symmetric matrix wrt a scalar

For a given square symmetric invertible matrix $\mathbf{X}$ and scalar $\alpha$ (such that the entries of $\mathbf{X}$ depend on $\alpha$), I would like to use the following well-known expression for ...
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### Derivative of the trace of $X^TP^TPX$ with respect to P

$\newcommand{\Tr}{\operatorname{Tr}}$ Consider the following expression: $\Tr(X^TP^TPX)$ where $X$ and $P$ are real matrices. What is the best way to approach the calculation of its derivative ...
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### Derivative of matrix product: is it true that $\frac{d}{dt}(A^TA) = 2A^T \frac{dA}{dt}$?

$A$ is a square matrix. All elements of $A$ depend on a parameter $t$, that is, $a_{ij}=a_{ij}(t)$. Let $S(A):=A^TA$, and take the derivative of $S$ w.r.t. $t$: $\displaystyle \frac{dS}{dt}$ Now, ...
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### Matricial differentiation $x x^{\top} b$

What is the drivative of $x x^{\top} b$ with respect to x, knowing that b is constant vector?
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### Matrix exponential Differentiation

We have the equation $e^X = \sum_{k=0}^\infty{1 \over k!}X^k.$, where X is a matrix of dimension $3 \times 3$ . Now I have a function $f(x)=C_1x+C_2*\frac{x^2}{2}$ where $C_1,C_2,f(x)$ has ...
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### total least squares derivation with matrices

Taken from a computer vision book: "to minimize the sum of the perpendicular distances between points and lines, we need to minimize $$\sum_i (ax_i + by_i +c)^2$$ subject to $a^2 +b^2 =1$. Now using ...
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### Derivative of $f(x)=\|Ax\|_2^2$

I'm trying to find the derivative of $f(x)=\|Ax\|_2^2$ where $A$ is some matrix and $\|u\|_2$ is the euclidean norm of $u$, $\|u\|_2 = \sqrt{u_1^2+u_2^2+\cdots+u_n^2}$ I know how to do this by ...
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### Element-wise derivative of the inverse of a matrix

I would appreciate if you could help me to obtain the element-wise derivative of $Z = (-A-BX)^{(-1)}$ where all of elements of $A$, $B$ and $X$ are positive. I conjecture that if I increase any of ...