T2: The Exponential Map Flashcards
Define the norm of an object
A function from k-vector space to +ve which defines a notion of length
Define the Jordan Cannonical form
Where a matrix can be expressed as an upper triangular matrix containing blocks along the diagonal. Each block is upper triangular with non-zero terms along the diagonal and ones along the super diagonal.
In what case is a Jordan cannonical form a diagonal matrix?
Where all the blocks have size 1
Under what criteria is the exponential map surjective?`
Over the complex domain
Define a one-parameter subgroup
A differentiable map from real numbers under addition, to the group GL_n(K). Equivalently, a group homomorphism between these groups.
What is the sufficient condition for f to be a one-parameter subgroup?
It is continuous and exists over some integral 0 to a.
Define the exponential map exp(X)
For X∈ gl_n (K), sum from 0 to infty X^k/k!
Gothic
Under what condition does exp(X+Y) = exp(X)exp(Y)
If X and Y commutes
Where is the exponential map uniformly convergent?
Over compact subsets of gl_n(K)
exp(sX) exp(tX) = ?
exp((s+t)X)
g exp(X) g^−1 = ?
exp(gXg^-1)
exp(tr(X)) = ?
det exp(X)
d/dt det(exp(tX)) at t=0 = ?
tr(X)
Define gl_n (K) (gothic)
= M_n(K) : the set of n x n matrices over K
Define the map t → exp(tX)
Map from reals to to GL_n(K)
