Find all the solutions of the following LP problemmaximize $z= 3x_1+x_2+0x_3$subject to$x_1+2x_2 \leq 5$$x_1+x_2-x_3 \leq 2$$7x_1 + 3x_2 -5x_3 \leq 20$and $x_1, x_2, x_3 \geq 0$my final LP is: $x_3 = 3-x_2-x_4+x_5$$x_1 = 5 - 2x_2 -x_4$$x_6 = 0 + 6x_2+2x_4+9x_5$$z = 15-5x_2-3x_4$I have found a solution of this LP $(5, 0, 3, 0, 0, 0)$, but I'm not sure what it means by finding 'all the solutions', can someone give me a hint?...Read more

Given that the system$$\dot x = f(x,y) \\ \dot y = g(x,y)$$is Hamiltonian $H(x,y)$.Introduce new coordinates $(z, w)$ defined by the linear transformation $z = ax + by\,$ and $w = cx + dy\,$ which we assume to be invertible. Show that thesystem in the new coordinates is also Hamiltonian, and compute the newHamiltonian $\widetilde{H}(z, w)$.My attempt: So I know that for a system to be a Hamiltonian it must be a fist integral, it must also obey that $\dot x = \frac{\partial H}{\partial y}$ and that $\dot y = -\frac{\partial H}{\partial x}$And I ...Read more

Consider a matrix $X\equiv\left[\begin{matrix}Z & W\end{matrix}\right]$, with $Z^{\intercal}P_{W}Z$ and $W^{\intercal}P_{Z}W$ nonsingular, where $P_{\bullet}\equiv \bullet(\bullet^{\intercal}\bullet)^{-1}\bullet^\intercal$. Intuitively, I'd say that $P_{X}P_{Z}=P_{Z}$ since $\mathrm{col}Z\subset \mathrm{colX}$, yet I can't show that from straight arithmetics. Next is my work.Denote $M_\bullet\equiv I-P_{\bullet}$. Then:$ P_{X}P_{Z}=\left[\begin{matrix}Z & W\end{matrix}\right]\left[\begin{matrix}Z^{\intercal}Z & Z^{\intercal}W\\W^{\i...Read more

Let $H_1,H_2\in\mathbb{Q}^{n\times n}$ be idempotent and symmetric matrices. For any $0<\mu<\frac{1}{2}$, consider the matrix$$H_\mu:=\mu H_1+(1-\mu)H_2.$$I'm looking for a description of $\text{Eig}(H_\mu,\mu)$.Clearly, $\text{img}(H_1)\cap\ker(H_2)\subseteq\text{Eig}(H_\mu,\mu)$, but under which conditions do we have equality? Is there a description of the "missing piece"$$\text{Eig}(H_\mu,\mu)/\text{img}(H_1)\cap\ker(H_2)?$$See also my question here....Read more

Given the upper Shift Matrix, which for e.g. dimension $5$ is$${\bf E}_{\,{\bf 5}} = \left( {\matrix{ 0 & 1 & 0 & 0 & 0 \cr 0 & 0 & 1 & 0 & 0 \cr 0 & 0 & 0 & 1 & 0 \cr 0 & 0 & 0 & 0 & 1 \cr 0 & 0 & 0 & 0 & 0 \cr } } \right)$$then its non-negative integral powers are just given by a shift of the non-null diagonal, till ${\bf E}_{\,{\bf 5}} ^{\,{\bf 5}} ={\bf 0}$.I know that the Jordan decomposition of ${\bf E}_{\,h} ^{\,{\bf n}}$ is given by $${...Read more

I have an expression involving matrices, of the form:$$f(k)=x^T A_k^{-1}A x$$ where $x$ is a $1\times N$ vector, $A_k = A + k I$ and $A$ is an $N\times N$ matrix ($A_k$ is invertible for all $k$) and $k>0$. It is known that $f(k)$ is real and monotonic increasing, but nothing more. I need to further analyze the behaviour of $f(k)$ and the simplest way would be to plugin my matrices and plot it. However, this becomes computationally intensive for large $N$ as it involves calculating the inverses repeatedly. One thought I had was to create a o...Read more

Given matrices $\Gamma_1, C \in \mathbb R^{n \times n}$, find a matrix $\Gamma \in \mathbb R^{n \times n}$ that minimizes the matrix norm of $\Gamma - \Gamma_1$ subject to constraints$$\begin{array}{ll} \text{minimize} & \| \Gamma - \Gamma_1 \|\\ \text{subject to} & \Gamma 1_n = 1_n\\ & \Gamma^{\top} 1_n = 1_n\\ & C \Gamma = 1_n 1_n^{\top}\end{array}$$How can I solve it using MATLAB? I find someone uses Iterative Bregman Projections. Or can this be solved by using proximal methods?Are there any methods to solve it. It's very imp...Read more

Let $ K $ be a field. We can recursively define matrices as$ M_{a} = (a)$ for any $ a\in K $ and$$ M_{a_1, \cdots, a_{2^i}} = \begin{pmatrix} M_{a_1, \cdots, a_{2^{i-1}}} & M_{a_{2^{i-1} +1}, \cdots, a_{2^i}}\\ M_{a_{2^{i-1} +1}, \cdots, a_{2^i}} & M_{a_1, \cdots, a_{2^{i-1}}}\\\end{pmatrix} $$when $ i>0 $ and $a_j\in K$.What is the name for the type of matrices?Let $ a_1, a_2, \cdots, a_{2^n} $ and$ b_1, b_2, \cdots, b_{2^n} $ be two list of elements in $ K $.Is there a formula for the eigenvalues of $$ M_{a_1, a_2, \cdots, ...Read more

A bisymmetric matrix is a square matrix that is symmetric about both of its main diagonals. If $A$ is a bisymmetric matrix and I'm interested in solving $Ax=b$. Are there techniques used to exploit this structure when solving the system of linear equations?Note: I'm looking for techniques which exploit more than just the fact that the matrix is symmetric....Read more

Are there special methods to get exact eigenvalues of a tridiagonal matrix?...Read more

One of the consequences of Seymour's characterization of regular matroids is the existence of a polynomial time recognition algorithm for totally unimodular matrices (i.e. matrices for which every square sub-determinant is in {0, 1, -1}).But has anyone actually implemented it?...Read more

I am working on an optimization problem where I have to find derivate of $⟨F(X),WF(X)Z⟩$ with respect to X. Here $⟨,⟩$is a standard inner product (Frobenius dot product), W & Z are a constant matrix, and $f$ a function of a matrix, whose output is also a matrix. I have previously asked a slightly different version of the same question, below is the solution for it \begin{align}\phi &= \langle F,WF \rangle = \langle W,FF^T \rangle\\\frac{\partial \phi}{\partial X} &= \langle W,dF F^T+FdF^T\rangle\\&=\langle W+W^T,dF F^T\rangle\...Read more

From Eq. 51 of the matrix cookbook we know that $\frac{\partial \log\det (AXB)}{\partial X} = (X^{-1})^\top$,where $\det(X)$ is the determinant of $X$.I was wondering what is the derivative of$\frac{\partial \log\det (AXB + C)}{\partial X}$. Is it still $ (X^{-1})^\top$?Thanks!...Read more

Hi,I'm attempting to implement a QR factorization with column pivoting so that the returned R matrix has decreasing diagonal elements (that is, $r_{i,i} \leq r_{i-1,i-1}$ for all $i\geq 2$). Mathematically, it would involve finding the matrix $P$ so that $AP=QR$, or $A=QRP^T$.I'm using Gram-Schmidt to compute QR.One source I found was this: http://www.mathworks.de/matlabcentral/newsreader/view_thread/250632Quote from the answer:"During the iteration #k, it is easy to show that if we pick among the set of remaining vectors (i.e., not yet include...Read more

You have a n by n matrix that increases in value going from left to right and from top to bottom. Here is an example matrix: 1 2 3 4 2 3 4 5 3 4 5 6 4 5 6 7 Now you have to find a "divide and conquer" algorithm to find if a value 'S'is in this sorted matrix. Find a recurrence relation and state the timecomplexity.Here is my solution:1. Find bottom left value (Time Steps: 1)2. If 'S' is greater than this value, move right, otherwise, move up3. Worse case, you traverse a whole row to the right or a whole column up. -So let this be notated as 'n...Read more