On the definition of symmetric matrices

In summary, the conversation discusses the difference between symmetric and Hermitian matrices, and their properties in relation to real and complex numbers. While Hermitian matrices have a complete set of eigenvectors and real eigenvalues, symmetric matrices can contain complex terms and do not have the same properties as Hermitian matrices. Hermitian matrices are considered more important due to their ability to satisfy the inner product equation with complex vectors.
  • #1
shakgoku
29
1
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.
 
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  • #2
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...
 
  • #3
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.
 
  • #4
For example, the matrix
[tex]\begin{bmatrix}1+ i & 2- 2i \\ 2- 2i & 3i\end{bmatrix}[/tex]
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
[tex]\begin{bmatrix}1+ i & 2- 2i \\ 2+ 2i & 3i\end{bmatrix}[/tex]
will have a complete set of eigenvectors.
 
  • #5
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.
 
  • #6
Of course. Thanks.
 
  • #7
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.
 

What is a symmetric matrix?

A symmetric matrix is a square matrix where the elements above and below the main diagonal are reflections of each other. In other words, the element at row i and column j is equal to the element at row j and column i.

How do you determine if a matrix is symmetric?

To determine if a matrix is symmetric, you can check if it is equal to its transpose. If the matrix is equal to its transpose, then it is symmetric.

Can a non-square matrix be symmetric?

No, a non-square matrix cannot be symmetric. Symmetry refers to the relationship between elements in a square matrix, so a non-square matrix does not have this property.

What are the properties of symmetric matrices?

Some properties of symmetric matrices include:
- They have real eigenvalues
- They are diagonalizable
- They have orthogonal eigenvectors
- They are always square

How are symmetric matrices used in real life?

Symmetric matrices are used in various fields of science and engineering, including physics, statistics, and computer science. They are commonly used to model and analyze systems with a high degree of symmetry, such as molecules, physical structures, and data sets. They are also useful in solving optimization problems and in machine learning algorithms.

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