Tagged Questions

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How do I find the differential of these functions?

We are given 2 functions: 1) $f: Mat_{nxn}(\mathbb R) \to Mat_{nxn} (\mathbb R)$ $f(A)=A^m$, $m>0$. and 2) $g: GL_n (\mathbb R) \to GL_n (\mathbb R)$ $g(A)=A^m$, $m<0$. Find the ...
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derivative of log(det(A)) wrt x, where A is matrix that depends on x

I have two large sparse matrices B and C, and I need to calculate $\frac{\rm{d}}{\rm{d}(\log({\lambda}) }\log( \det(B+\lambda C))$. Because B and C are very large I can't directly evaluate the ...
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Gradient of a scalar function with respect to a matrix

I need to calculate $\dfrac{\partial}{\partial K}f(K)$, with: $$f(K)=-\frac{1}{2}(u-Kx)^T\Sigma^{-1}(u-Kx)$$ $K$ and $\Sigma$ are $n\times n$ matrices, $\Sigma$ is symmetric, $u$ and $x$ are column ...
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Extremum of a multidimensional quadratic function

I have the following function: $$g(h) = h'\Sigma\Sigma'h-h'm-r,$$ where $h$ is a vector in $\mathbb{R}^M$, $\Sigma$ is a $M\times K$ matrix such that $\Sigma\Sigma'$ is positive definite and has ...
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How to max $f(D)$ over the space where matrix $D$ is diagonal?

I want to maximize some function $f(D).$ Obviously if there is no constraints, I can just form matrix $G$ by $G(D)_{ij} = \frac{\partial f(D)}{\partial D_{ij}}$ and solve $G(D) = 0$ for D. ...
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Say I have a function $(A+B)^{-1}$ where $A$, $B$ are matrix-valued functions of some vector $x$. Can I then expand this function around $x=0$ as: $$(A+B)^{-1} = (A[0]+B[0])^{-1} - (A[0]+B[0])^{-2} ... 1answer 109 views For A(t) differentiable, taking positive matrices as values, how show \sqrt{A(t)} is differentiable? On p. 150 of Lax's Linear Algebra, he mentions that is is not hard to show that if R(t) is a differentiable matrix-valued function of a single variable, whose values are positive matrices, then the ... 0answers 105 views least squares: verify simple derivative What is the derivative of ||Y-X\beta||_2 w.r.t X? I have (Y-X\beta)^T(Y-X \beta) = Y^TY-Y^TX \beta-\beta^TX^TY+\beta^TX^TX\beta which gives the derivative as -Y\beta^T -Y\beta^T ... 1answer 116 views Having trouble using eigenvectors to solve differential equations The question asked to solve$$\frac{dx}{dy} = \begin{pmatrix} 5 & 4 \\ -1 & 1\\ \end{pmatrix}x$$,where$$ x = \begin{pmatrix} x_1 \\ x_2 \\ \end{pmatrix} I went ...
How can you describe all $2\times 2$ matrices whose eigenvalues are 0 and 1? My attempt: I know that 0 and 1 has to be solutions of the characteristic polynomial. And I've considered some examples ...
Let's say you have a two-state markovian source whose transition matrix is $P=\begin{pmatrix}1-\sigma & \sigma\\ \tau & 1-\tau\end{pmatrix}$, for the state 0 the data rate is 0 and for the ...