matrices - How to find all solutions of a given LP problem

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

matrices - Given an initial system is a Hamiltonian. Show that a linear transformation of this system is also a Hamiltonian and find what the Hamiltonian is

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

matrices - Projection onto submatrix, projected onto matrix

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

convexity - Eigenspace of convex combination of two idempotent matrices

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

matrices - Jordan decomposition of powers of the Shift Matrix

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

matrices - Constructing equivalent algebraic expressions for matrix equations

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

matrices - Using iterative projection to solve a minimization problem

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

What is the name for the type of matrices?

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

matrices - Bisymmetric Matrix, solving set of linear equations.

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

multivariable calculus - Derivative of inner product of function of matrices

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

matrices - QR factorization: How to get decreasing r_ii

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: 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

matrices - Is this a proper algorithm/time complexity for finding a value in this matrix?

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