(a) Show that every diagonal entry of a positive definite...

(a) Show that every diagonal entry of a positive definite matrix must be positive.
(b) Write down a symmetric matrix with all positive diagonal entries that is not positive definite. (c) Find a nonzero matrix with one or more zero diagonal entries that is positive semi-definite.

Key Concepts

Positive Definite Matrix

A positive definite matrix is a symmetric matrix A for which the quadratic form x?Ax is greater than zero for every nonzero vector x. This fundamental property ensures that all eigenvalues of A are positive and that the quadratic form is strictly convex. In the context of the question, using a standard basis vector (which has one nonzero entry) directly shows that the corresponding diagonal entry of A must be positive.

Quadratic Forms

Quadratic forms are expressions of the type x?Ax, where A is a symmetric matrix and x is a vector. They are instrumental in connecting the entries of a matrix to the matrix’s definiteness properties. Evaluating the quadratic form on basis vectors reveals the sign of the diagonal entries, and forms the basis for proving properties about positive definiteness or semi-definiteness.

Positive Semi-Definite Matrix

A positive semi-definite matrix is a symmetric matrix for which the quadratic form x?Ax is non-negative for all vectors x, allowing for the possibility that the result is zero even for some nonzero x. This definition broadens the concept of positive definiteness and is useful in finding examples of matrices that might have zero diagonal entries but still do not produce negative values when evaluated in a quadratic form.

Symmetric Matrices

Symmetric matrices are matrices that equal their own transpose (A = A?). This property is fundamental in linear algebra because it guarantees that the eigenvalues are real and facilitates the analysis of quadratic forms, which is crucial when determining positive definiteness or semi-definiteness. Symmetry is a key requirement in the standard definitions of positive definite and positive semi-definite matrices.

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