) Apply Warshall's algorithm to find transitive closure of the following digraph. Give its djacency matrix first. Then list the resulting matrix R for each intermediate step. b d go
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- Orthogonal linear operator : has the standard basis matrix: Find an orthonormal basis in which the operator matrix has a canonical form and write out this matrix. Specify the axis and angle of rotation defined by the operatorH.W: find value of matrix A eigen -2 3 A = 2.3. () Apply Warshall's algorithm to find transitive closure of the following digraph. Give its adjacency matrix first. Then list the resulting matrix R for each intermediate step. d
- Find the eigenvalues and eigenvectors of the given matrix. - 30 5 6 - 1 The eigenvalue(s) is/are (Simplify your answer. Use a comma to separate answers as needed.) The eigenvector(s) is/are. (Use a comma to separate vectors as needed.)(Cayley-IHamilton Theorem) The Cayley-Ilamilton theorem is a powerful theorem in Linear algebra that states: every square matrix of real numbers satisfies its own char- acteristic equation. For 2 x 2 matrices this can be seen in the following way. First, we introduce the trace of the matrix tr A 41,1 +22 and the determinant det A = a11a22 - 41,22,1- Then, the chacteristic polynomial is the quadratic given by p(A) = x – (tr A)A + (det. A). Then, for any 2 x 2 matrix, if you plug it into this polynomial, you should always get a zero matrix. c_h_test Function: Input parameters: • a single 2 x2 matrix for which you would like to test the Cayley Ilamilton Theorem Local Variables: • two scalars to represent the trace and determinant Output parameters: a single 2 x2 which is the result of evaluating the charcteristic polynomial at A (note: think carefully about the constant term) A possible sample case is: > mat_B = c_h_test([3, 2 ; 2, 0]) mat B = 0 0) Assume A is k × n-matrix and Q is n x n-invertible matrix. Prove that rank(AQ) = rank(A).
- #1 In this sequence of problems, we will generate random matrices (2d arrays), and calculate the product by using by both inner and outer product methods and then compare them with the product implemented in numpy. Verify that a matrix multiplied by a column vector is simply a linear combination of the columns of the matrix. Namely, if the columns of the matrix A are a_1, a_2, .. a_n and the vector x is (x_1,... x_n), then Ax = x_1 a_1 + ... + x_n a_n. Explain why this is simply the outer product way of multiplying A and x.6. Using the root vectors, work out the weight diagram of 10.H.Wi find eigen value and eigen vector of matrix A. A = 7 10 %3D -2