For a square matrix , the matrix exponential is defined bywhere is the identity matrix.
- [(1)] Let Compute the matrix exponential .
- [(2)] Consider the matrices where is a real number, and define the vector Let . Determine the minimum and maximum values of over all nonzero real vectors .
- [(3)] Let where is the matrix from part (2). Find the value of for the vector satisfying Let be the eigenvalues of the matrix in part {\rm (1)}. Let and be the minimum and maximum values, respectively, obtained in part {\rm (2)}, and letbe the value obtained in part {\rm (3)}. Compute\
[THOUGHT] I will:
- Compute the matrix exponential exp(A) and eigenvalues of A.
- Compute C = B*exp(A_x), verify its eigenvalues, get m (minimum eigenvalue).
- Solve C z = a symbolically to find z, then compute R^2 = z^T z and simplify.
- Form the final expression λ1λ2λ3 * m * R^2 * (λ1^2+λ2^2+λ3^2) using symbolic eigenvalue invariants (det and trace of A^2).
- Simplify to a final exact value.
Now I run the calculations in Python. [/THOUGHT] [PYTHON] import sympy as sp
Part (1): define A and compute exp(A) and eigenvalues
A = sp.Matrix([[-1,4,4], [-5,8,10], [3,-3,-5]])
A_eigvals = A.eigenvals() print("Eigenvalues of A with multiplicities:", A_eigvals)
A_exp = A.exp() print("exp(A) =") sp.pprint(A_exp)
Compute eigenvalue invariants:
detA = A.det() traceA2 = (A**2).trace() print("det(A) =", detA) print("trace(A^2) =", traceA2)
Part (2): define Ax, B, and e^{Ax}, then C = B*e^{Ax}
x = sp.symbols('x', real=True) Ax = sp.Matrix([[0, -x, 0], [x, 0, 0], [0, 0, 1]])
B = sp.diag(1, -1, 1)
Exponential of Ax: we know the analytic form, but also verify via series if desired.
I'll construct it directly using the rotation/e part:
c = sp.cos(x) s = sp.sin(x) Ax_exp = sp.Matrix([[c, -s, 0], [s, c, 0], [0, 0, sp.E]]) print("exp(Ax) =") sp.pprint(Ax_exp)
C = B * Ax_exp print("C = B*exp(Ax) =") sp.pprint(C)
print("Is C symmetric?", C.equals(C.T)) print("Eigenvalues of C:", C.eigenvals())
From theory, y^T C y / (y^T y) has min/max = min/max eigenvalues of C since C is symmetric.
So:
C_eigs = [ev for ev in C.eigenvals().keys()] m_val = sp.min(C_eigs) M_val = sp.max(C_eigs) print("m (min) =", m_val) print("M (max) =", M_val)
Part (3): solve Cz = a, then R^2 = z^T z
a_vec = sp.Matrix([1,1,sp.E]) z1, z2, z3 = sp.symbols('z1 z2 z3', real=True) z = sp.Matrix([z1,z2,z3])
Solve C z = a for z
sol = sp.solve(C*z - a_vec, (z1,z2,z3), dict=True) print("Solution for z:", sol)
z_sol = sp.Matrix([sol[0][z1], sol[0][z2], sol[0][z3]]) print("z vector =", z_sol)
R_sq = sp.simplify((z_sol.T * z_sol)[0]) print("R^2 simplified =", R_sq)
Now compute the final expression:
lam_prod = detA # λ1 λ2 λ3 lam_sq_sum = traceA2 # λ1^2 + λ2^2 + λ3^2 final_expr = sp.simplify(lam_prod * m_val * R_sq * lam_sq_sum) print("Final expression =", final_expr) [/PYTHON]
[THOUGHT] Using the determinant of (A), the trace of (A^2), the minimal Rayleigh quotient eigenvalue (m), and the squared norm (R^2) of the critical point, the product evaluates to [ \lambda_1\lambda_2\lambda_3 \cdot m \cdot R^2 \cdot (\lambda_1^2+\lambda_2^2+\lambda_3^2) = -6 \cdot (-1) \cdot 3 \cdot 14 = 252. ] [/THOUGHT] \boxed{252}