Inner product of complex numbrs

[DmytriE]

I would like if someone could either verify or clarify my thinking about inner products.

There is a matrix, V that is m x n, that is made up of complex numbers. When matrix V is multiplied by its hermitian then the product is a matrix with the same integer down the main diagonal (i.e. Eigenvalues are all the same).

 

[0xDEADBEEF]

If you mean by its Hermitian the conjugate transpose. There are matrices that are self adjoined, and don't change under that operation. When you multiply such a matrix by its conjugate transpose the eigenvalues are squared.

 

[DmytriE]

What do you mean by self adjoining?

 

[0xDEADBEEF]

The definition of a self adjoint or Hermitian matrix is, that the matrix is the same after mirroring on the main diagonal and complex conjugation. What you mean by Hermitian of a matrix I don't know.

 

[alphysicist]

The inner product of two complex numbers, denoted as <a,b>, is defined as <a,b> = a* x b, where a* represents the complex conjugate of a. This definition holds true for matrices as well. When two matrices are multiplied together, their inner product is equal to the sum of the products of each corresponding element in the matrices, with the complex conjugate of the first matrix element multiplied by the second matrix element. In other words, the inner product of two matrices is a complex number that represents the measure of similarity between the two matrices.

In the context of matrix V and its hermitian, the inner product of V and its hermitian can be represented as <V,V*>. This inner product will result in a matrix with integer values along the main diagonal, as stated in the content. This is because the complex conjugates of each element in V will cancel out, leaving only the real parts of the complex numbers to be multiplied together.

It is important to note that the inner product of complex numbers and matrices is a generalization of the dot product in Euclidean space. Just like the dot product, the inner product of complex numbers and matrices also follows the properties of linearity, commutativity, and positive definiteness.

In summary, the inner product of complex numbers and matrices is a useful tool in measuring the similarity between two entities. It is defined as the sum of the products of each corresponding element, with the complex conjugate of the first element multiplied by the second element. In the case of matrix V and its hermitian, the inner product results in a matrix with integer values along the main diagonal.