(a) Prove that a diagonal matrix D=diag(c1, c2, …, cn) is positive definite if and only if all i

step 1 For this exercise, we have that D sub 1 can equal 3 .002. And we have S equal to 1 over root 5,

(a) Prove that a diagonal matrix D=diag(c1, c2, …, cn) is...

(a) Prove that a diagonal matrix $D=\operatorname{diag}\left(c_1, c_2, \ldots, c_n\right)$ is positive definite if and only if all its diagonal entries are positive: $c_i>0 . \quad(b)$ Write down and identify the associated inner product.

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Key Concepts

Positive Definiteness

A matrix is positive definite if, for any nonzero vector x, the quadratic form x?Ax is strictly positive. This concept is fundamental in many areas such as optimization and stability analysis as it guarantees unique minimizers and well-behaved systems.

Diagonal Matrix

A diagonal matrix is a matrix in which the entries outside the main diagonal are all zero. In the context of positive definiteness, a diagonal matrix simplifies the analysis because the quadratic form reduces to a weighted sum of squares, with the weights given by the diagonal entries.

Quadratic Forms

Quadratic forms are expressions of the form x?Ax, where A is a symmetric matrix and x is a vector. They are closely related to the study of positive definiteness because the sign of the quadratic form determines whether a matrix is positive definite, semidefinite, or indefinite.

Inner Product

An inner product is a generalization of the dot product that defines geometric concepts like length and angle in vector spaces. When associated with a positive definite matrix A, the inner product can be defined as ?x, y? = x?Ay, which in the diagonal case yields a weighted sum of the products of the vector components.

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