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Minimize quadratic form
The document discusses the derivative of a quadratic form Q(x) = xTAx. It shows that the gradient of Q(x) is equal to 2Ax. To find the saddle point of Q(x), the derivative with respect to x, which is 2Ax, is set equal to 0. Solving this results in an expression for the coordinates x and y of the saddle point in terms of the coefficients of the quadratic form matrix A.
Minimize quadratic form
- 1. Derivative of quadraticform 2016 May 28 (Saturday) Hanpen Robot AM 7:07 (Japan time)
- 2. 𝑄 𝕩 =𝕩 𝑡 𝔸𝕩 = 𝑥 𝑦 𝑎 ℎ ℎ 𝑏 𝑥 𝑦 𝑄 𝕩 = 𝑥 𝑦 𝑎 ℎ ℎ 𝑏 𝑥 𝑦 = 𝑥 𝑦 𝑎𝑥 + ℎ𝑦 ℎ𝑥 + 𝑏𝑦 ∴ 𝑄 𝕩 = 𝑎𝑥2 + 2ℎ𝑥𝑦 + 𝑏𝑦2 𝜕 𝜕𝕩 : = 𝜕 𝜕𝑥 𝜕 𝜕𝑦 T , 𝑇ℎ𝑎𝑡 𝑖𝑠 𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡.
- 3. 𝜕𝑄 𝜕𝕩 = 𝜕𝑄 𝜕𝑥 𝜕𝑄 𝜕𝑦 T = 2𝑎𝑥 + 2ℎ𝑦 2ℎ𝑥+ 2𝑏𝑦 = 2 𝑎 ℎ ℎ 𝑏 𝑥 𝑦 ∴ 𝜕 𝕩 𝑡 𝔸𝕩 𝜕𝕩 = 2𝔸𝕩
- 4. Find a quadricform 𝑄(𝕩) saddle point. 𝜕𝕩 𝑡 𝔸𝕩 𝜕𝕩 = 2𝔸𝕩 = 2 𝑎 ℎ ℎ 𝑏 𝑥 𝑦 = 0 0 ∴ 𝑥 𝑦 = 1 𝑎𝑏 − ℎ2 𝑏 −ℎ −ℎ 𝑎 0 0 = 0 0 (𝑎𝑏 − ℎ2 ≠ 0)