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- (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 0PYTHON PROGRAMMING/ LU FACTORIZATION (Please provide the python code for the solution)Q4: Write the parametric equation of revolution surface in matrix form only which generated by rotate a Bezier curve defined by the coefficient parameter in one plane only, for the x-axis [0,5, 10,4], y-axis [1,4,2,2] respectively, for u-0.5 and 0 = 45° Note: [the rotation about y-axis].
- 6. (Linear independence ): Is it true that if x; y; and z are linearly independent vectors overGF(q), then so are x + y; y + z, and z + x ?2. Matrix Quantum Mechanics. Use python matrix and algebraic tools for this problem. Consider orthogonal vectors [1], [2) and [3). The operators C and P are represented by C = /C₁ 00 0 C₂ 0 0 0 C3. /P₁ 0 0 P = 0 0 P2 0 P3 Where all quantities are real. a) Do the operators C and P commute? b) Find the eigenvalues and normalized eigenvectors of the given operators.Determined the eigenvector of the matrix: please print A=[1 0 4, 0 4 0, 3 5 -3]
- 3 : Prove that the sum of the three variables of the sum exponent f(x) = Σe is strictly convex. For this i=1 purpose, compute the Hessian matrix of the function f(x).Please solve quickly, Matlab programming languageQuestion 5. For each of the following matrices, perform two iterations of the power method by hand starting with a vector of all ones. State the resulting approximations of the eigenvalue and eigenvector. $$ \begin{array}{1} A=\left[\begin{array}{11} 1 & 2 \\ 3 & 4 \end{array}\right] \\ B=\left[\begin{array}{ccc} 2 & -1 & 0 \ -1 & 2 & -1 \\ O & -1 & 2 \end{array}\right] \end{array} cs. VS. 1598| $$