Thread: 2x2 matrix - help needed!

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Hello,

Originally Posted by JrShohin

A matrix is positive definite if its eigenvalues are all positive.

Originally Posted by Moo

Hello,

A matrix is positive definite if its eigenvalues are all positive.

That's all?

Originally Posted by JrShohin

That's all?

Well, if you know how to find the characteristic polynomial, then that's all

The roots of the characteristic polynomial are the eigenvalues. So find such that there are indeed positive.

Characteristic polynomial - Wikipedia, the free encyclopedia
Positive-definite matrix - Wikipedia, the free encyclopedia (but I'm afraid this article may not correspond to your level...I can't tell)

Let M be an n × n [...] matrix. The following properties are equivalent to M being positive definite:
1. All eigenvalues λi of M are positive. [...]

That's the part you need.

Originally Posted by Moo

Well, if you know how to find the characteristic polynomial, then that's all

The roots of the characteristic polynomial are the eigenvalues. So find such that there are indeed positive.

Characteristic polynomial - Wikipedia, the free encyclopedia
Positive-definite matrix - Wikipedia, the free encyclopedia (but I'm afraid this article may not correspond to your level...I can't tell)

That's the part you need.

Thank you,

to tell the truth, i'm just beginning linear algebra, and plus, i haven't worked on maths for a loooong time and i forgot...

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Originally Posted by JrShohin

And how about this last one?

Hmmm

A diagonal 2x2 matrix is in the form
And the equivalent diagonal matrix to a given matrix will be formed by its eigenvalues. That is to say, in our case here, and are the eigenvalues of A.

Did you find them ?

For the eigenvectors, it's better you have a look on the internet...
Try this : Pauls Online Notes : Differential Equations - Review : Eigenvalues & Eigenvectors (it's also explain how to get the eigenvalues)
Take the time to read it please