On the definition of symmetric matrices

[shakgoku]

Can a symmetric matrix contain complex elements(terms).
If no, how is it that 'eigen values of a symmetric matrix are always real'(from a theorem)

Is a symmetric matrix containing complex terms called a hermitian matrix or is there any difference?

Can we call the following matrix symmetric (A = transpose of A), even though its not hermitian. (A not equal to A dagger)

i i
i 2i

But its eigen values are not real contradicting the above theorem.

 

[micromass]

The notion of a symmetric matrix does makes sense for complex matrices. So the matrix you mention can be called a complex symmetric matrix. Unfortunately, a lot a of beautiful theorems that hold for real symmetric matrices, fail to hold for complex symmetric matrices (for example: the theorem you mention). So that's why there is so little interest for symmetric matrices.

On the other hand, the hermitian matrices do share a lot of properties with the real symmetric matrices. And they are therefore much more interesting.

So, if somebody talks about symmetric matrices, then they are almost always real. If somebody wants to discuss complex matrices, then they will almost always use hermitian matrices instead of symmetric matrices...

 

[AlephZero]

A symmetric matrix has a_ij = a_ji for all i and j
A Hermitian matrix has a_ij = a*_ji for all i and j.

Complex symmetric and complex Hermitian matrices are different.
Real symmetric and real Hermitian matrices are the same, since a_ij = a*_ij if a is a real number.

I would prefer to put it the other way round from what Micromass said. There are many useful and interesting theorems about Hermitian matrices (and especially about Hermitian positive definite matrices). There is almost nothing extra that depends on a matrix being real and symmetric, as well as Hermitian.

Complex symmetric matrices do occur in some physics situations, for example mechanical vibrations including damping and analysing electrical circuits with alternating current, but they don't have the same "nice" properties as Hermitian matrices.

 

[HallsofIvy]

For example, the matrix
$$\begin{bmatrix}1+ i & 2- 2i \\ 2- 2i & 3i\end{bmatrix}$$
is "symmetric" but does not have the properties a real symmetric matrix would have (real eigenvalues and a complete set of eigenvectors for example).

A Hermitian matrix
$$\begin{bmatrix}1+ i & 2- 2i \\ 2+ 2i & 3i\end{bmatrix}$$
will have a complete set of eigenvectors.

 

[AlephZero]

HOI, that's not a Hermitian matrix. The diagonal terms have to be real, to make a_11 = a*_11, etc.

Your general comment is true, of course.

 

[HallsofIvy]

Of course. Thanks.

 

[adriank]

The point of a symmetric real matrix A is that <Ax, y> = <x, Ay> for any vectors x and y. (Here <,> is the usual inner product, i.e. dot product.)

This equation doesn't hold for complex symmetric matrices and complex vectors, but it does hold for Hermitian matrices, and that's what makes Hermitian matrices more important than complex symmetric matrices.