Equivalent Conditions
If U is a square, complex matrix, then the following conditions are equivalent:
- U is unitary
- U * is unitary
- U is invertible, with U –1=U *.
- the columns of U form an orthonormal basis of with respect to the usual inner product
- the rows of U form an orthonormal basis of with respect to the usual inner product
- U is an isometry with respect to the usual norm
- U is a normal matrix with eigenvalues lying on the unit circle.
Read more about this topic: Unitary Matrix
Famous quotes containing the words equivalent and/or conditions:
“I started off rapping for people just like myself, people who were in awe of wealth and flash. It was a conversation between me and them. But now most of those who buy my records are listening in on others’ conversation. They are the aural equivalent of voyeurs, thrilled at this crazy world that has nothing to do with their experience.”
—Ice-T [Tracy Marrow], U.S. rap musician. Observer (London, Oct. 27, 1991)
“We have got onto slippery ice where there is no friction and so in a certain sense the conditions are ideal, but also, just because of that, we are unable to walk. We want to walk so we need friction. Back to the rough ground!”
—Ludwig Wittgenstein (1889–1951)