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) 8 O b) 9 c) 24 d) 18
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- 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?a. Build an adjacency matrix ? for this map. b. How many paths of length 2 from V5 to V1 exist? c. How many paths of length 3 from V5 to V1 exist?If there is a non-singular matrix P such as P-1AP=D, matrix A is called a diagonalizable matrix. A, n x n square matrix is diagonalizable if and only if matrix A has n linearly independent eigenvectors. In this case, the diagonal elements of the diagonal matrix D are the eigenvalues of the matrix A. A=({{1, -1, -1}, {1, 3, 1}, {-3, 1, -1}}) : 1 -1 -1 1 3 1 -3 1 -1 a)Write a program that calculates the eigenvalues and eigenvectors of matrix A using NumPy. b)Write the program that determines whether the D matrix is diagonal by calculating the D matrix, using NumPy. #UsePython
- 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 +…Consider the matrix multiplication C = AB. A matrix W is called a witness matrix with the following conditions: • If ci j = 0, then wi j = 0.• If ci j = 1, then wi j = k where aik = bkj = 1. make code to displays two Booelan matrices A and B and their productC. Note that we define these matrices with 0 and 1 entries and declare their typesas boolean with dtype = bool statement after specifying the entries. Alternatively,we could have specified theses matrices with True and False entries. The booleanproduct C is then converted to integer by the astype method and both forms of Care printedA 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…
- 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.Let f ∈ C+ 2π with a zero of order 2p at z. Let r>p and m = n/r. Then there exists a constant c > 0 independent of n such that for all nsufficiently large, all eigenvalues of the preconditioned matrix C−1 n (Km,2r ∗ f)Tn(f) are larger than c./*** We will assume that `matrixA` and `matrixB` are valid 2D arrays of* `int`s. Each matrix is rectangular, with each row having the same number* of columns.** First, we need to make sure the matrices are compatible. Given `numRows X* numColumns` for each matrix `A` and `B`, if `A` is `m x n`, then `B` must* be `n x p`. That is the ``inner`` dimensions (lengths) must match.** If number of columns in `matrixA` does NOT equal the number of rows in* `matrixB`, then return `null`. That is, not output is created.** Assuming the dimensions are consistent, then we create a 2D output array* that is `m x p`, that is `numRowsA X numColumnsB`.** To calculate each element of the output matrix, we multiply the rows of* `A` by the columns of `B` and sum them up.** For example, let for integer values be given by lower case letters, let* `A` be a 2 x 3 matrix, and `B` a 3 x 3 matrix. The inner dimensions are* consistent, so we can multiply.** <pre>[a, b ,c]A = [d, e, f][p, q, r]B = |s, t, u|[v,…
- Write a program that can inverse a matrix by using an approach to find its minor, cofactor and adjugate matrix. Test your program with the given matrix ?Give a clear description of an efficient algorithm for finding the k smallest elements of a very large n-element vector. Compare its running time with that of other plausible ways of achieving the same result, including that of applying k times your solution for part (a). [Note that in part (a) the result of the function consists of one element, whereas here it consists of k elements. As above, you may assume for simplicity that all the elements of the vector are different.]The Algebraic AlgorithmThe algebraic BFS algorithm uses the adjacency matrix A of a graph G = (V, E).We first initialize a vector x which consists of all zeros except the index of the sourcevertex s that we want to run the algorithm from; we then form the matrix A = A+ Iand then AT · x selects all nodes which are at distance 1 (level 1) from the sourcevertex. Multiplying the vector x with the matrix A2 gives the vertices that are at most2 hops. In general, the product Ak · x will give neighbors that are at most k hopsaway and the multiplication should be performed boolean as in Algorithm 7.2.Algorithm 7.2 Algebraic BFS1: Input : Adjacency matrix An,n of a graph G = (V, E) connected, unweighted graph G and asource vertex s2: Output : N, visited, levels a matrix that shows level i vertices at its column i, the visitedvertices in sequence and their levels3: x[n] ← 04: x[s] ← 15: A ← A + I6: for i = 1 to n do7: N ← AT · x8: levels[i] ← column i of N9: visited ← visited∪ {visited nodes…