Prova Algebra Linear

Sustituto en expresión\nA = P D P^T (A es simétrica y P ortogonal)\n\n\\[ x^T A x = x^T P D P^T x = \\]\n\\[ y_j^T D_{y_j} \\]\n\ncon \\( P x = y \\) luego \\( x = P y \\)\n\n\\[ x^T A x = x^T P D P^T x = 3 y_1^2 + y_2^2 + y_3^2 = 48 \\]\ncon \\( x = P y \\)\n\nD:\n\\[ D = \\begin{bmatrix} 3 & 0 \\\\ 0 & 7 \\end{bmatrix} \\]\n\nP:\n\\[ P = \\begin{bmatrix} \\frac{1}{\\sqrt{2}} & \\frac{1}{\\sqrt{2}} \\\\ \\frac{1}{\\sqrt{2}} & -\\frac{1}{\\sqrt{2}} \\end{bmatrix} \\]\n\nP^T:\n\\[ P^T = \\begin{bmatrix} \\frac{1}{\\sqrt{2}} & \\frac{1}{\\sqrt{2}} \\\\ \\frac{1}{\\sqrt{2}} & -\\frac{1}{\\sqrt{2}} \\end{bmatrix} \\]\n\nA = P D P^T\n\n\\[ A = \\begin{bmatrix} \\frac{3}{2} & \\frac{3}{2} \\\\ \\frac{1}{2} & \\frac{1}{2} \\end{bmatrix} \\]\n\n\\[ \\begin{bmatrix} 3 & -2 \\\\ -2 & 5 \\end{bmatrix} = \\begin{bmatrix}5 & -2 \\\\ -2 & 5 \\end{bmatrix}\\]