Suppose we have three points in a two dimensional space: (3,1), (2,1), and (7,7). We want to perform PCA on these points, so we construct a 2-by-2 matrix whose eigenvectors are the directions that best represent these three points. Construct this matrix, as outlined in the PCA algorithm discussed in class, and identify, in the list below, one of its elements. O a) 25.5 O b) 14 O c) 27 O d) 12

Suppose we have three points in a two dimensional space: (3,1), (2,1), and (7,7). We want to perform PCA on these points, so we construct a 2-by-2 matrix whose eigenvectors are the directions that best represent these three points. Construct this matrix, as outlined in the PCA algorithm discussed in class, and identify, in the list below, one of its elements. O a) 25.5 O b) 14 O c) 27 O d) 12

Operations Research : Applications and Algorithms

4th Edition

ISBN:9780534380588

Author:Wayne L. Winston

Publisher:Wayne L. Winston

Chapter2: Basic Linear Algebra

Section2.4: Linear Independence And Linear Dependence

Problem 8P

See similar textbooks

Similar questions

Create the matrix of the graph above and answer the following questions (If there isno arrow at the end of both sides of line, it is bidirectional).Answer the following questions by examining the matrix and its powers, show yourresults step by step clearly:(a) How many walks of length 2 are there from 2 to 4?(b) How many walks of length 2 are there from 3 to 5?(c) How many walks of length 3 are there from 0 to 5?(d) How many walks of length 3 are there from 2 to 4?
Let y be a column vector: y = [1, 2, 3, 4, 5, 6] so that y.shape = (6,1). Reshape the vector into a matrix z using the numpy.array.reshape and (numpy.array.transpose if necessary) to form a new matrix z whose first column is [1, 2, 3], and whose second column is [4, 5, 6]. Print out the resulting array z.
Consider a vector ('L', 'M', 'M', 'L', 'H', 'M', 'H', 'H'), where H stands for ‘high’, L stands for ‘low’, and M for ‘medium’ in rstudio. a. Convert this vector into an unordered factor. b. Convert this vector into an ordered factor with Low < Medium < High.
Type in Latex **Problem**. Let $$A = \begin{bmatrix} .5 & .2 & .3 \\ .3 & .8 & .3 \\ .2 & 0 & .4 \end{bmatrix}.$$ This matrix is an example of a **stochastic matrix**: its column sums are all equal to 1. The vectors $$\mathbf{v}_1 = \begin{bmatrix} .3 \\ .6 \\ .1 \end{bmatrix}, \mathbf{v}_2 = \begin{bmatrix} 1 \\ -3 \\ 2 \end{bmatrix}, \mathbf{v}_3 = \begin{bmatrix} -1 \\ 0 \\ 1\end{bmatrix}$$ are all eigenvectors of $A$. * Compute $\left[\begin{array}{rrr} 1 & 1 & 1 \end{array}\right]\cdot\mathbf{x}_0$ and deduce that $c_1 = 1$.* Finally, let $\mathbf{x}_k = A^k \mathbf{x}_0$. Show that $\mathbf{x}_k \longrightarrow \mathbf{v}_1$ as $k$ goes to infinity. (The vector $\mathbf{v}_1$ is called a **steady-state vector** for $A.$) **Solution**. To prove that $c_1 = 1$, we first left-multiply both sides of the above equation by $[1 \, 1\, 1]$ and then simplify both sides:$$\begin{aligned}[1 \, 1\, 1]\mathbf{x}_0 &= [1 \, 1\, 1](c_1\mathbf{v}_1 +…
%Encode A1, b1 and x1 as the vector of unknowns. A1 = b1 = syms xv1 = %Check the size of A, set it as m1 and n1 [m1,n1] = %Augment A and b to form AM1 AM1 = %Solve the Reduced Rwo Echelon of AM1. RREFA1 = %Collect the last column and set as bnew1, set the remaining elements as Anew1 bnew1= Anew1 = %Check if Anew is an identity matrix, if it is, bnew is the solution if Anew1 =eye(m1,n1) Root1 = bnew1 else display("No Solution") end %Augment the matrix A1 with the identity Matrix of the same size, set the result as AMI1 AMI1 = %Find the reduced row echelon form of AMI1, set the result as RREFAI1 RREFAI1= %Collect the second half of the matrix as AInew1, set the remaining elements as AIold1 AInew1= AIold1 = %Check if AIold1 is an identity matrix, if it is, AInew1 is the inverse %Encode A2, b2 and xv2 as the vector of unknowns. A2 = b2 = syms xv2 = %Check the size of A2, set it as m2 and n2 [m2,n2] = %Augment A2 and b2 to form AM2 AM2 = %Solve the…
Consider the un-directed graph shown in the Fig. The values inside the node refer to the feature value of the node. The order of the nodes is A,B,C,D when following the matrix form in below questions.(You can use python or other languages to find the eigen values and other operationfor this question):1. For the given graph, write down the degree matrix, adjacency matrix, and Laplacian matrix representation.2.Find the eigenvalues and eigenvectors of the graph Laplacian matrix. Do you see any pattern pertaining to the signs of the eigenvalues (all positive or negative or no such pattern). Will it be same for all such graph Laplacianmatrices, and why?
Problem 1) Change the diagonal elements of a 3X3 matrix to 1. Diagonal elements of the example matrix shown here are 10, 8, 2 10 -4 0 7 8 3 0 0 2 Read the given values from input (cin) using for loops and then write your code to solve the diagonal.
write a C++ program to Given a matrix of dimension m*n where each cell in the matrix can have values 0, 1 or 2 whichhas the following meaning:0: Empty cell1: Cells have fresh oranges2: Cells have rotten orangesSo we have to determine what is the minimum time required so that all the oranges becomerotten. A rotten orange at index [i,j] can rot other fresh orange at indexes [i-1,j], [i+1,j], [i,j-1],[i,j+1] (up, down, left and right). If it is impossible to rot every orange then simply return -1.Examples:Input: arr[][C] = { {2, 1, 0, 2, 1},{1, 0, 1, 2, 1},{1, 0, 0, 2, 1}};Output:All oranges can become rotten in 2 time frames.Input: arr[][C] = { {2, 1, 0, 2, 1},Tahir Iqbal Department of Computer Sciences. BULC{0, 0, 1, 2, 1},{1, 0, 0, 2, 1}};Output:All oranges cannot be rotten.Below is algorithm.1) Create an empty Q.2) Find all rotten oranges and enqueue them to Q. Also enqueuea delimiter to indicate beginning of next time frame.3) While Q is not empty do following3.a) While delimiter in…
In Computer Science a Graph is represented using an adjacency matrix. Ismatrix is a square matrix whose dimension is the total number of vertices.The following example shows the graphical representation of a graph with 5 vertices, its matrixof adjacency, degree of entry and exit of each vertex, that is, the total number ofarrows that enter or leave each vertex (verify in the image) and the loops of the graph, that issay the vertices that connect with themselvesTo program it, use Object Oriented Programming concepts (Classes, objects, attributes, methods), it can be in Java or in Python.-Declare a constant V with value 5-Declare a variable called Graph that is a VxV matrix of integers-Define a MENU procedure with the following textGRAPHS1. Create Graph2.Show Graph3. Adjacency between pairs4.Input degree5.Output degree6.Loops0.exit-Validate MENU so that it receives only valid options (from 0 to 6), otherwise send an error message and repeat the reading-Make the MENU call in the main…
A matrix is an array of numbers of size m by n (i.e., m x n). When we multiply 2 matrices, we multiply the matching numbers, then sum them up. Multiplying a matrix of size m x n with another matrix of size n x q will result in a matrix of size m x q. An example is shown in Figure 1: [ ? ? ? ? ? ? ] x [ ? ? ? ? ] = [ ?? +?? ?? + ?? ?? + ?? ?? + ?? ?? + ?? ?? + ?? ] Matrix 3 x 2 Matrix 2 x 2 Matrix 3 x 2 Figure 1: Multiplication of Matrix 3 x 2 and 2 x 2 Write a C++ program that multiplies 2 matrices and return the sum of the transposed matrix. Your program shall include the following: a) Define two 2 dimensional arrays, M and N that each represents a matrix of size 3 x 2 and a matrix of size 2 x 2. Request the user to enter the values for the 2 arrays using advanced pointer notations (refer to your slides for the list of array pointer notations). b) Create a function named multMatrix() that passes as arguments, the values of array M and N, multiplies them and store the results in a new…
#plea# Count the number of unique paths from a[0][0] to a[m-1][n-1]# We are allowed to move either right or down from a cell in the matrix.# Approaches-# (i) Recursion- Recurse starting from a[m-1][n-1], upwards and leftwards,# add the path count of both recursions and return count.# (ii) Dynamic Programming- Start from a[0][0].Store the count in a count# matrix. Return count[m-1][n-1]# T(n)- O(mn), S(n)- O(mn)# def count_paths(m, n): if m < 1 or n < 1: return -1 count = [[None for j in range(n)] for i in range(m)] # Taking care of the edge cases- matrix of size 1xn or mx1 for i in range(n): count[0][i] = 1 for j in range(m): count[j][0] = 1 for i in range(1, m): for j in range(1, n): # Number of ways to reach a[i][j] = number of ways to reach # a[i-1][j] + a[i][j-1] count[i][j] = count[i - 1][j] + count[i][j - 1]…
Imagine you have a square matrix, where each cell (pixel) is either black or white. Design an algorithm to find the maximum subsquare such that all four borders are filled with black pixels.