Why is the transpose of the matrix of cofactors of A called the adjoint of A? Does it adjoin something?
While searching for this on the web I noticed something called self-adjoint. What?s that?
I wish textbooks would tell you this stuff, elliptic intergrals were a mystery to me until I found out why they were called that.
Thanks
Self-adjoint means that A[sup]T[/sup] = A.
I don’t know what it’s called adjoint, though.
Actually, self-adjoint means A[sup][/sup] = A, where "[sup][/sup]" denotes the conjugate of the transpose of A (also known as the adjoint of A). A matrix which satisfies A[sup]T[/sup]=A is usually just called symmetric.
I’ve never heard of the term “adjoint” being applied to the matrix of cofactors.
This site’s pretty cool, although it won’t entirely answer your question :
Earliest known uses of some of the words of mathematics
Note that there’s two different adjoints in use – the transpose of the matrix of cofactors is referred to in some places as the classical adjoint. The other one is called the Hermitian adjoint to distinguish it, but it’s usually plain ‘adjoint’ refers to it.
The (Hermitian) adjoint of a matrix A is the transpose of the conjugate matrix of A. The conjugate matrix of A is the matrix in which each element is replaced by it’s complex conjugate (remember complex numbers? (a+bi)* = (a-bi)).
Self-adjoint only refers to this type of adjoint, but does mean that the matrix equals its adjoint. Self-adjoint matrices are also called Hermitian.
Now it appears even my symbols are screwing things up here, so I’m not sure if Giraffe meant T to be transpose or as a dagger, which has been used to symbolize the adjoint.
Often the adjoint is signified by *. What makes it more confusing is that I just used * to indicate the complex conjugate. If * is used for adjoint, a bar over the top indicates complex conjugation, so there’s no confusion.
So if A is Hermitian, A = A* = (\A)[sup]T[/sup] (where you have to imagine that \A means I’ve drawn a line over A).
In the alternate notation, A = A[sup]t[/sup] = (A*)[sup]T[/sup]. (where you have to imagine that [sup]t[/sup] is a dagger).
If A[sup]T[/sup] means transpose, it’s only true that A = A[sup]T[/sup]=> A is self-adjoint if A has only real values.
Thanks. My textbook only talks about
Unitary matrices
A = (\A)[sup]T[/sup] (panamajack notation)
Hermitian matrices
A = A*
Adn Normal matrices
AA = A*A = I
It would have saved me alot of time if they would have told me the other names for this stuff, but I guess they can’t cover everything.
Thanks again for your help
It wasn’t a typo – I was just trying to give a quick, simple answer, relevant to the OP. I assumed he was dealing strictly with real numbers, so that adjoint and transpose are equivalent operations.
Then you thorough bastards showed up and made me look bad…