Orthogonal and symmetric matrices

[mnb96]

Hello,
I guess this is a basic question.
Let´s say that If I am given a matrix X it is possible to form a symmetric matrix by computing $X+X^{T}$.

But how can I form a matrix which is both symmetric and orthogonal? That is:
$$M=M^{T}=M^{-1}$$.

 

[marcusl]

You have implicitly stated that M is real. In this case I think only the identity matrix matches your requirements.

 

[chingkui]

And also diagonal matrix with 1 or -1 at diagonal. Any more?

 

[mnb96]

Thanks for the answers.
I just noticed that unfortunately I stated my problem incorrectly.

Starting from a matrix, I wanted to find another matrix which is symmetric (not Hermitian!) and unitary. That is:

$$M=M^{T}$$
$$MM^\dagger=I$$

Here $$M^{T}$$ means "transpose", while $$M^\dagger$$ means "conjugate transpose".

 

[blue_raver22]

Hello,

Yes, this is a great question! To form a matrix that is both symmetric and orthogonal, we can use the Gram-Schmidt process. This process takes a set of linearly independent vectors and orthonormalizes them, meaning that they are all perpendicular to each other and have a length of 1. This can be used to create an orthogonal matrix, and then we can simply multiply it by its transpose to get a symmetric matrix.

Another way to form a symmetric and orthogonal matrix is by using the QR decomposition. This method decomposes a matrix into an orthogonal matrix (Q) and an upper triangular matrix (R). By setting R to be a diagonal matrix with the same values as Q, we can create a matrix that is both symmetric and orthogonal.

I hope this helps! Let me know if you have any other questions.