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sodaboy7
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Eigen values of a complex symmetric matrix which is NOT a hermitian are not always real. I want to formulate conditions for which eigen values of a complex symmetric matrix (which is not hermitian) are real.
A symmetric matrix is hermitian iff the matrix is real, so that is not a good way to characterize symmetric complex matrices.Robert1986 said:That is, I am saying that a symmetric matrix is hermitian iff all eigenvalues are real.
AlephZero said:A symmetric matrix is hermitian iff the matrix is real, so that is not a good way to characterize symmetric complex matrices.
I don't think there is a simple answer to the OP's question.
Robert1986 said:So, what I am saying is that there are no complex symmetric matrices with all real eigenvalues. (Unless, of course, the matrix is real.)
AlephZero said:OK, I agree with that.
But the number of complex eigenvalues can by anything from 1 to the order of the matrix, which doesn't go very far to answer the OP's question.
AlephZero said:Maybe we both misunderstood, but I read the OP's "which eigen values ... are real" as a question about some of them, not all of them.
It might be possible to give an answer if the eigenproblem represents a physical system. For example the eigenvalues of a damped multi-degree-of-freedom oscillator, with an arbitrary damping matrix, represent the damped natural freuqencies on the s-plane, therefore they are all complex except for zero-frequency (rigid body motion) modes. Also, the sign of the real part of the eigenvalues shows whether the mode is damped, undamped, or unstable (i.e. it gains energy from outside the system).
But I don't know how to turn that "physics insight" about a particular physical system into a mathematical way to characaterize the matrix.
sodaboy7 said:What I am trying to say is this.
All hermitian matrices are symmetric but all symmetric matrices are not hermitian. Eigenvalues of hermitian (real or complex) matrices are always real. But what if the matrix is complex and symmetric but not hermitian. In hermitian the ij element is complex conjugal of ji element. But I am taking about matrix for which ij element and ji element are equal. Eigen values of such a matrix may not be real. So under what condition Eigenvalues will be real.
But it has not been proved in this thread (nor is a reference to a proof given) that a symmetric matrix must be diagonalizable.Robert1986 said:I should have been more clear. Any symmetric matrix [itex]M[/itex] has an eigenbasis (because any symmetric matrix is diagonalisable.)
But it is not proved, at this stage, that [itex]x^*Mx[/itex] is real for all ##x##, even if ##M## is symmetric and diagonalizable with all eigenvalues real. In that case, this holds if ##x## is an eigenvector to ##M##, but since we don't know that the eigenbasis is orthogonal, this cannot be generalized to all ##x##.Robert1986 said:EDIT:
In fact, given any matrix [itex]M[/itex], if [itex]x^*M^*x[/itex] is real for all [itex]x[/itex] then [itex]M[/itex] is hermitian.
Yes, and as I think about it, there are really simple counterexamples to what I said.Erland said:But it has not been proved in this thread (nor is a reference to a proof given) that a symmetric matrix must be diagonalizable.
The eigenvalues of a complex symmetric matrix are the special set of numbers that, when multiplied by the matrix, result in a scalar multiple of the original matrix. In other words, the eigenvalues represent the "stretching" or "shrinking" factors of the matrix when it is transformed.
The eigenvalues of a complex symmetric matrix can be calculated by solving the characteristic equation, which is obtained by subtracting the identity matrix multiplied by a scalar from the original matrix and setting the determinant of the resulting matrix equal to 0.
Yes, a complex symmetric matrix can have multiple eigenvalues. This means that there can be more than one "stretching" or "shrinking" factor when the matrix is transformed.
The eigenvalues of a complex symmetric matrix have many applications in mathematics, physics, and engineering. They are used to solve systems of differential equations, analyze the stability of physical systems, and perform data compression, among other things.
Yes, there is a direct relationship between the eigenvalues and eigenvectors of a complex symmetric matrix. Each eigenvalue has a corresponding eigenvector, which represents the direction in which the matrix is transformed when multiplied by that eigenvalue.