(a) For which numbers b (allowing both positive and negative...

(a) For which numbers $b$ (allowing both positive and negative numbers) is the matrix $A=\left(\begin{array}{ll}1 & b \\ b & 4\end{array}\right)$ positive definite? (b) Find the factorization $A=L D L^T$ when $b$ is in the range for positive definiteness. (c) Find the minimum value (depending on $b$; it might be finite or it might be $-\infty)$ of the function $p(x, y)=x^2+2 b x y+4 y^2-2 y$.

Key Concepts

Positive Definiteness

Positive definiteness is a property of a symmetric matrix where all its eigenvalues are strictly positive. This is typically checked by ensuring that all the leading principal minors of the matrix are positive, a method known as Sylvester's criterion. Positive definiteness guarantees that the quadratic form associated with the matrix is strictly convex, which implies the existence of a unique global minimum in optimization problems.

LDL^T Factorization

LDL^T factorization is a matrix decomposition method for symmetric matrices, in which a matrix is expressed as the product of a lower triangular matrix L, a diagonal matrix D, and the transpose of L. This factorization is particularly useful in numerical linear algebra because it avoids the need for square roots, can improve computational stability, and is often used in algorithms for solving systems of equations or for simplifying quadratic forms.

Quadratic Forms Optimization

Quadratic forms optimization involves finding the minimum value of a quadratic function, which can be expressed in the form x^T A x + b^T x + c. The nature of the quadratic form, particularly the definiteness of the matrix A, is crucial in determining whether the function has a finite minimum or if it is unbounded. Techniques such as completing the square or analyzing the eigenvalues of A are typically used to derive conditions for the existence of a minimum and to compute its value.

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