2.1 Definition and identification of critical points

Critical points are where a function's gradient vanishes, marking potential extrema or saddle points. Understanding these points is crucial for analyzing function behavior and solving optimization problems on manifolds.

The Hessian matrix, containing second-order partial derivatives, helps classify non-degenerate critical points. It determines whether a point is a local minimum, maximum, or saddle point, providing key insights into function topology.

Critical Points and Gradients

Definition of Critical Points and Gradients

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• Critical point is a point $p$ on a smooth function $f$ where the gradient $\nabla f(p) = 0$
• Gradient $\nabla f$ is a vector that points in the direction of the greatest rate of increase of the function $f$ and its magnitude is the rate of change in that direction
• Smooth function is a function that has continuous derivatives up to some desired order over some domain
• Manifold is a topological space that locally resembles Euclidean space near each point (examples include curves, surfaces, and higher-dimensional spaces)

Properties of Critical Points

• Critical points can be classified as non-degenerate or degenerate based on the behavior of the function around the point
• Non-degenerate critical points have a well-defined type (local minimum, local maximum, or saddle point) determined by the second derivative test
• Degenerate critical points do not have a well-defined type and require higher-order derivatives to classify
• Critical points play a crucial role in optimization problems and understanding the behavior of functions on manifolds

Classification of Critical Points

Types of Non-Degenerate Critical Points

• Non-degenerate critical point is a critical point where the Hessian matrix (matrix of second partial derivatives) is invertible
• Local maximum is a non-degenerate critical point where the function values in a small neighborhood around the point are less than or equal to the value at the point (example: the peak of a hill)
• Local minimum is a non-degenerate critical point where the function values in a small neighborhood around the point are greater than or equal to the value at the point (example: the bottom of a valley)
• Saddle point is a non-degenerate critical point that is neither a local maximum nor a local minimum (example: a mountain pass between two peaks)

Degenerate Critical Points

• Degenerate critical point is a critical point where the Hessian matrix is not invertible
• Classification of degenerate critical points requires examining higher-order derivatives
• Degenerate critical points can exhibit more complex behavior than non-degenerate critical points (example: a monkey saddle, which has three valleys meeting at a point)

Hessian Matrix

Definition and Properties of the Hessian Matrix

• Hessian matrix is a square matrix of second-order partial derivatives of a scalar-valued function
• For a function $f(x_1, \ldots, x_n)$, the Hessian matrix $H(f)$ is defined as:

\frac{\partial^2 f}{\partial x_1^2} & \frac{\partial^2 f}{\partial x_1 \partial x_2} & \cdots & \frac{\partial^2 f}{\partial x_1 \partial x_n} \\ \frac{\partial^2 f}{\partial x_2 \partial x_1} & \frac{\partial^2 f}{\partial x_2^2} & \cdots & \frac{\partial^2 f}{\partial x_2 \partial x_n} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial^2 f}{\partial x_n \partial x_1} & \frac{\partial^2 f}{\partial x_n \partial x_2} & \cdots & \frac{\partial^2 f}{\partial x_n^2} \end{bmatrix}$$ - The Hessian matrix is symmetric if the second partial derivatives are continuous ### Applications of the Hessian Matrix - The Hessian matrix is used to classify non-degenerate critical points using the second derivative test - If the Hessian matrix is positive definite at a critical point, the point is a local minimum - If the Hessian matrix is negative definite at a critical point, the point is a local maximum - If the Hessian matrix has both positive and negative eigenvalues at a critical point, the point is a saddle point - The determinant of the Hessian matrix can be used to determine whether a critical point is non-degenerate (non-zero determinant) or degenerate (zero determinant)