# Eigenvalues, Curvature, and Quadratic Forms

This application shows how the shape of a quadratic form is controlled by the eigenvalues of its matrix. For this application the matrix is diagonal and so its eigenvalues are simply the elements on the diagonal.

Identifier:
http://demonstrations.wolfram.com/EigenvaluesCurvatureAndQuadraticForms/
Rating:
Creator(s):
Chris Maes
Cataloger:
Tom Polaski
Publisher:
Wolfram Demonstrations Project
Rights:
Chris Maes and Wolfram Research
Format Other:
Requires downloading the CDF player from Wolfram (https://www.wolfram.com/cdf-player/).

## Comments

### notion of curvature of quadratic form is used but not defined

The user selects two eigenvalues by selecting a point in the square $-1<x<1$, $-1<y<1$. The quadratic form $\mathbf{x}'D\mathbf{x}$ is graphed, where $D$ is the diagonal matrix whose entries are the selected eigenvalues. From the graph, one sees the connection between the eigenvalues and positive (negative) definite quadratic forms. The notion of the curvature of a quadratic form is used but not defined.

### User can control the two eigenvalues

Shows the curvature of a quadratic form for a $2 \times 2$ matrix. User can control the two eigenvalues which range between -1 and 1.