Hessian Matrix\Max Min Analysis, Eigenvalues etc

In summary, in calc 3 class, the Hessian Matrix is being used to find eigenvalues and determine the nature of points (max, min, saddle, or indeterminant) in multivariable equations. The teacher's explanation may be difficult to understand due to a thick accent, and there is limited coverage of this topic in the textbook. The use of Hessian matrices can be further explored through resources such as Wikipedia, which provides information on critical points, discriminant, and the second derivative test. Additional helpful resources may include textbooks or other sources for better understanding and preparation for upcoming evaluations.
  • #1
alec_tronn
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In my calc 3 class, we've taken an alternative(?) route to learning maxes and mins of multivariable equations. By using a Hessian Matrix, we're supposed to be able to find the eigenvalues of a function at the point, and determine whether the point is a max, min, saddle point, or indeterminant. Also, using these eigan values, a new axis system is formed. I can only vaguely understand most of what the teacher has explained (thick middle-eastern accent), and I have until Thursday to fully understand everything. Our book does not cover the matrix ways of doing things.

Does anyone have any useful links on hessian matrices? I have looked, but they all seem to go deeper into matrix things or use notation that I'm not familiar with.

Any books I should try to find (textbooks or otherwise)? Everybody got a D on the first quiz, and I'm not looking forward to that happening again on the test. Thanks for any information, links, or explanations that you can provide.
 
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  • #2
The Hessian matrix is just the name of the matrix (read "table") in which the second order derivatives are stored. Just as the gradient vector (while written in matrix notation) is the table in which the first order derivatives are stored.

See http://en.wikipedia.org/wiki/Hessian_matrix ans particularly the "Critical points and discriminant" and "Second derivative test" paragraphs.
 

1. What is a Hessian matrix?

A Hessian matrix is a square matrix of second-order partial derivatives of a multivariable function. It is used in mathematical optimization to determine the critical points and classify them as maxima, minima, or saddle points.

2. How is a Hessian matrix used in max-min analysis?

In max-min analysis, the Hessian matrix is used to determine the nature of the critical points of a multivariable function. The eigenvalues of the Hessian matrix are used to classify the critical points as maxima, minima, or saddle points.

3. What are eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are concepts from linear algebra that are used in the analysis of matrices. Eigenvalues are scalar values that represent the scaling factor of the eigenvectors when they are multiplied by a matrix. Eigenvectors are non-zero vectors that remain in the same direction after being multiplied by a matrix.

4. How are eigenvalues and eigenvectors used in the analysis of a Hessian matrix?

In the analysis of a Hessian matrix, the eigenvalues and eigenvectors are used to determine the nature of the critical points. If all eigenvalues are positive, the critical point is a local minimum. If all eigenvalues are negative, the critical point is a local maximum. If there are both positive and negative eigenvalues, the critical point is a saddle point.

5. Can the Hessian matrix be used in all types of functions?

No, the Hessian matrix can only be used in functions that are twice continuously differentiable. This means that the function must have continuous first and second-order partial derivatives. If the function does not meet this criteria, the Hessian matrix cannot be used for max-min analysis.

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