Recent content by ali987

Thanks every one for their useful comments, but, what about the main question? We have a matrix (with real entries) whose all eigenvalues lie on the unit circle and that matrix is not orthogonal. What other conditions may be exist for this matrix? Or in other words, how can we proof that all...

Unfortunately, that matrix is not unitary(or Orthogonal, since all entries are real). A unitary matrix has eigenvalues on the unit circle, but if all eigenvalues of a matrix lie on the unit circle, that matrix is not definitely unitary. that's the problem.

I can't talk about it definitely, but, according to a numerical simulation results, I had several identical eigenvalues. so, there are less than "n" unique and different eigenvalues. But, as is mentioned, I can't extend this conclusion to general case, these are results of just a numerical...

It is definitely invertible. All entries are real, but eigenvalues can be complex.

Dear all, I have a matrix, namely A. I calculate its eigenvalues by MATLAB and all of its eigenvalues lie on the unit circle(their amplitudes equal to 1). But A is not an orthogonal matrix (transpose(A) is not equal to inverse(A) ). What other condition or relationship may be correct for it?

Unfortunately, there is no special condition on A. B1, B2, B3 and B4 are constructed from several matrices themselves, some of those matrices are symmetric and positive definite. Is there any theorem which relate orthogonality of the A to orthogonality (or sth like that) of B1...B4?

Hi everyone Consider a 2x2 partitioned matrix as follow: A = [ B1 B2 ; B3 B4 ] I'm sure that all eigenvalues of A are on the unit circle (i.e., abs (all eig) = 1 ). but, I don't know how to prove it. Is there any theorem?

\begin{array}{l} {X_1} = A_2^{ - 1}{B_2} \\ {X_2} = \Delta A_2^{ - 1}L \\ {X_3} = \Delta A_1^{ - 1}{L^T} \\ {X_4} = A_1^{ - 1}{B_1} \\ \end{array} In which \begin{array}{l} {A_1} = G + \frac{{{\Delta ^2}}}{4}{L^T}{C^{ - 1}}L \\ {A_2} = C + \frac{{{\Delta ^2}}}{4}L{G^{ - 1}}{L^T} \\...

Hi everyone, consider two following partitioned matrices: \begin{array}{l} {M_1} = \left[ {\begin{array}{*{20}{c}} { - \frac{1}{2}{X_1}} & {{X_2}} \\ {{X_3}} & { - \frac{1}{2}{X_4}} \\ \end{array}} \right] \\ {M_2} = \left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}}...