|WikiProject Mathematics||(Rated Start-class, Mid-importance)|
In the last section, Uses, some confusion arises regarding the different meanings of "unitarily equivalent" and "unitarily similar".
I am fairly confident that the spectral theorem states that a normal matrix is unitarily equivalent to a diagonal matrix, with unitarily equivalent defined as follows:
A square matrix A is considered unitarily equivalent to a matrix B if there exists a unitary matrix U that satisfies A=UBU^\dagger, where U^\dagger means taking the complex conjugate of the transpose of U.
Now, an mxn matrix A is considered unitarily similar to an mxn matrix B if there exist two unitary matrices U (nxn) and T (mxm) satisfying A=TBU^\dagger. This definition plays a role in the cited singular value decomposition theorem.
So I'd prefer to see the two terms swap places, if people can agree on using the words with the meaning outlined above.
- I think the definitions as used in the literature are slightly confusing. Two matrices A and B are similar if A = XBX-1 for some matrix X, and they are unitarily equivalent if A = XBX-1 for some unitary matrix X. Sometimes, the more logical phrase unitarily similar is used for unitarily equivalent. Now comes the confusing bit: A and B are equivalent if A = XBY-1 for some matrices X and Y. The term equivalent is not used very often, because two matrices are equivalent iff they have equal rank. Reference: Horn and Johnson, Matrix Analysis; but also see similarity (mathematics).
- I edited the article to clarify the definition of unitarily equivalent. Jitse Niesen 11:59, 30 Mar 2005 (UTC)
In the opening, it says, "Thus, the matrix D = (di,j) with n columns and n rows is diagonal if:", but I think it should say "if and only if", otherwise it's not making a strong enough claim. Can someone confirm this? If so, please make the change. —Preceding unsigned comment added by 126.96.36.199 (talk) 14:16, 11 March 2009 (UTC)