What Does the Book Say About the Eigenvalues of 3x3 Matrices?

[LagrangeEuler]

I found in one book that every quadratic matrix 3x3 has at least one eigenvalue. I do not understand. Shouldn't be stated at least one real eigenvalue? Thanks for the answer.

 

[fresh_42]

I found in one book that every quadratic matrix 3x3 has at least one eigenvalue. I do not understand. Shouldn't be stated at least one real eigenvalue? Thanks for the answer.

Yes. I assume that the book is primarily assuming real matrices.

We get a characteristic polynomial which decomposes into linear factors in case of an algebraic closed field. So we have $\chi(t)=-(t-\lambda_1)(t-\lambda_2)(t-\lambda_3)$. But we do not have any knowledge whether the algebraic multiplicities are all one. E.g. $\lambda_1=\lambda_2=\lambda_3$ cannot be ruled out, what commonly is called one eigenvalue, not three.

 

[martinbn]

By definition eigenvalues have to belong to the base field. A qubic polynomial with real quoeficients always has a real root.

 

[LagrangeEuler]

Yes. But if we have complex 3x3 matrix is it possible to have only one eigenvalue?

 

[PeroK]

Yes. But if we have complex 3x3 matrix is it possible to have only one eigenvalue?

Like the diagonal matrix $diag(i, i, i)$?

 

[martinbn]

Yes. But if we have complex 3x3 matrix is it possible to have only one eigenvalue?

And what is the book talking about? May be if you revieled the title and the page or quoted the book, we wouldn't have to guess.