- #1

swampwiz

- 571

- 83

But I'd to prove this singleness from the opposite direction. I know that the sesquilinear quadratic product for the case of unitary matrices as the side matrix is such that the normality of the product is the same as the normality of the central matrix, and thus there must exist a unitary matrix that diagonalizes any particular normal matrix since the sesquilinear product of a diagonal matrix and a unitary matrix (namely the inverse of the original one, which is the transpose because it is unitary, and thus unitary itself) must be normal to follow the normality of a diagonal matrix (which is de facto normal). But I can't seem to prove that there only exists 1 unitary matrix that accomplishes this.

EDIT: Each column can be

__+__, so there are 2

^{n}modal matrices, but if the columns are limited such that the element that corresponds to the pivot element is the same sign, then this situation goes away.

So I am hung up as to why there is only 1 solution (unless I am wrong about this!) for the unitary diagonalization, and as well the singleness of the eigenvector solution. Obviously, once it is proven that there is only 1 for each, then it is proven that this matrix is one in the same.