Diagonalisability problem and others

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Homework Help Overview

The discussion revolves around two problems involving linear algebra concepts, specifically focusing on the properties of matrices and vector norms. The first problem addresses the diagonalizability of the product of a matrix and its transpose, while the second problem involves maximizing and minimizing a quadratic form associated with a given matrix on a unit sphere.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore the properties of the matrix product AtA, with one suggesting that proving its symmetry may be a starting point. Questions arise regarding the dimensions of the vector x in the second problem and its implications for the calculations involved.

Discussion Status

The discussion is active, with participants providing hints and clarifications regarding the properties of matrices and the requirements for the vectors involved. Some guidance has been offered, particularly about the nature of the vector x in relation to the matrix A.

Contextual Notes

There is a mention of potential confusion regarding the dimensions of the matrix and vector involved in the second problem, which may affect the interpretation of the problem. Additionally, the original poster expresses uncertainty about the properties needed to prove the diagonalizability in the first problem.

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Homework Statement


1.Prove that if A is a real matrix then At A is diagonalisable.

2. Given a known 3*3 matrix A, Calculate the maximum and minimum values of ||Ax|| on the sphere ||x|| = 1.


Homework Equations





The Attempt at a Solution


For the first problem, I'm thinking of proving that AtA is symmetric, but I'm not sure which properties to use.
For the second one, is X a 3*n matrix? Do I need to discuss n?
Any help is greatly appreciated!
 
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rainwyz0706 said:
For the first problem, I'm thinking of proving that AtA is symmetric, but I'm not sure which properties to use.
If [itex]B=A^TA[/itex], write down what [itex]B_{ij}[/itex] and [itex]B_{ji}[/itex] are in terms of the elements of A and show that they're equal.
 
Thanks a lot! I can't believe I missed it in the first place.
Could anyone give me some hints about problem 2?
 
For the second one, yes, x must be such a vector that the product Ax is meaningful, i.e. a 3x1 vector (matrix).
 
thx, I finished it!
 

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