Finding eigenvalues of a 3x3 matrix

I am having problems trying to orthogonally diagonalise the below matrix. I am stuck at the stage where you find the eigenvalues.

The matrix is

5 -4 0

-4 3 -4 = 0

0 -4 1

This is what I get, but I cannot solve it so I think I have done something wrong.

5-λ -4 0

-4 3-λ -4 = 0

0 -4 1-λ

(5-λ)*|(3-λ)(1-λ) - (-4)(-4)| + 4*|(-4)(1-λ)| = 0

this works down to

(5-λ)(λ^2 +12λ - 29)

when typing this I have just realised that I just need to use the quadratic equation to solve (λ^2 +12λ - 29) and (5 - λ) is just λ = 5.

So if someone could just confirm if I'm heading towards the right direction or not, then that would be great (Happy)

=

Re: Finding eigenvalues of a 3x3 matrix

Hi marky82! :)

I think you made a calculation error.

In (5-λ)*|(3-λ)(1-λ) - (-4)(-4)| + 4*|(-4)(1-λ)| = 0, you dropped a factor (5-λ) in the 2nd term.

Furthermore (5-λ) is not a factor of the entire expression.

You are heading in the right direction though.

Re: Finding eigenvalues of a 3x3 matrix

thanks for replying. I am not sure what you mean. Could you show me how you worked out the eigenvalues so I can compare it with mine please?

Re: Finding eigenvalues of a 3x3 matrix

Well, you can see the eigenvalues for instance here.

As you can see 5 is not one of them.

Re: Finding eigenvalues of a 3x3 matrix

Quote:

Originally Posted by

**ILikeSerena** Hi marky82! :)

I think you made a calculation error.

In (5-λ)*|(3-λ)(1-λ) - (-4)(-4)| + 4*|(-4)(1-λ)| = 0, you dropped a factor (5-λ) in the 2nd term.

Furthermore (5-λ) is not a factor of the entire expression.

You are heading in the right direction though.

Thanks, for the link, it helps when I have answer to work towards. My maths terminology isn't the greatest so I don't really understand the above quote. I don't get what it means to have dropped a factor (5 - λ) in the 2nd term. It's probably so simple but my brain doesn't get what you mean :(

Re: Finding eigenvalues of a 3x3 matrix

Apologies, It has just clicked. Thanks so much for your help :)

Re: Finding eigenvalues of a 3x3 matrix

Aaah! You mean the meaning of the words I've used!?

Are you referring to *term *and *factor*?

Or something else?

I'd like to know for future reference.

Re: Finding eigenvalues of a 3x3 matrix

Quote:

Originally Posted by

**ILikeSerena** Aaah! You mean the meaning of the words I've used!?

Are you referring to *term *and *factor*?

Or something else?

I'd like to know for future reference.

yeah, term and factor. I normally find I can do most maths processes but the terminology is what I have to spend most time on getting to grips with.

Re: Finding eigenvalues of a 3x3 matrix

Quote:

Originally Posted by

**marky82** yeah, term and factor. I normally find I can do most maths processes but the terminology is what I have to spend most time on getting to grips with.

Yeah, I know what you mean.

Whenever I learn something new, the first challenge is always to come to grips with the new words that are being used, or otherwise with the words that I simply did not know (well enough) yet.

Re: Finding eigenvalues of a 3x3 matrix

Yeah its like its starting to sound like another language when I speak it out loud.