Question

If \(A\) is positive definite, show that each diagonal entry is positive.

Short answer

Each diagonal entry of a positive definite matrix is positive because the quadratic form \( e_i^T A e_i > 0 \).

Step-by-step solution

Define Positive Definiteness

A matrix "\( A\) is said to be positive definite if for any non-zero vector \( x\), the quadratic form \( x^T A x \) is positive, i.e., \( x^T A x > 0 \). This property will be used to show the positivity of diagonal elements.

Consider the Canonical Basis Vectors

Let \( e_i \) be the standard basis vector in \( \mathbb{R}^n \) whose \( i \)-th component is 1, and all other components are 0.

Compute the Quadratic Form with Basis Vectors

For the matrix \( A \) and the basis vector \( e_i \), compute the quadratic form: \( e_i^T A e_i \). This equates to the \( i \)-th diagonal entry of \( A \).

Apply Positive Definite Property

Since \( A \) is positive definite, \( e_i^T A e_i > 0 \) for each basis vector \( e_i \). Therefore, \( A_{ii} > 0 \), which implies that every diagonal entry \( A_{ii}\) of \( A \) is positive.

Key concepts

Quadratic Form

A quadratic form is an expression involving a symmetric matrix and a vector, and it is essential in understanding positive definite matrices. The quadratic form for a vector \( x \) with respect to a matrix \( A \) is given by \( x^T A x \).

Here's a breakdown of the components:

• \( x^T \) is the transpose of vector \( x \).
• \( A \) is a symmetric matrix.
• \( x \) is the same vector.

The result of this operation is a scalar value. For a matrix to be considered positive definite, this quadratic form must be strictly positive for all non-zero vectors \( x \).

This concept is crucial when checking the positivity of diagonal elements in such matrices because it offers a way to mathematically prove that certain elements of the matrix meet the required conditions. By focusing the quadratic form on canonical basis vectors, we can simplify the test to individual matrix components.

Canonical Basis Vectors

Canonical basis vectors, also known as standard basis vectors, are vital tools for inspecting matrices. These vectors are denoted as \( e_i \), where the vector has a 1 in the \( i \)-th position and 0s elsewhere.

They serve as the building blocks of any vector space, facilitating straightforward analyses of matrices. For any matrix \( A \), the action of using a canonical basis vector \( e_i \) is profound because:

• The quadratic form \( e_i^T A e_i \) essentially picks out the \( i \)-th diagonal element \( A_{ii} \) of the matrix \( A \).
• This approach allows one to isolate and evaluate specific elements efficiently without the need for complex calculations.

Using these vectors is an effective method to pinpoint specific properties, such as checking whether each individual diagonal element of a positive definite matrix is positive.

Diagonal Elements

The diagonal elements of a matrix, often denoted \( A_{ii} \), are the entries positioned along the line that stretches from the top-left to bottom-right of the matrix. In a square matrix \( A \), these elements are crucial in determining properties like positive definiteness.

Here’s how they relate:

• In the context of a positive definite matrix, each diagonal element must be positive.
• This condition can be demonstrated by leveraging the quadratic form \( e_i^T A e_i \), focusing on each basis vector \( e_i \).
• The positivity of the quadratic form implies \( A_{ii} > 0 \) for each \( i \), confirming that all diagonal elements are indeed positive.

Understanding the properties of diagonal elements helps in comprehending the overall behavior of the matrix, making them a central focus when dealing with linear transformations and stability assessments in various mathematical fields.