Key Sequence
- we began the chapter defining \(T^m\) (reminding ourselves the usual rules of \(T^{m+n} = T^{m}T^{n}\), \((T^{m})^{n} = T^{mn}\), and, for invertible maps, \(T^{-m} = (T^{-1})^{m}\)) and \(p(T)\), wrapping copies of \(T\) into coefficients of a polynomial, and from those definitions showed that polynomial of operator is commutative
- we then used those results + fundamental theorem of algebra to show that operators on complex vector spaces have an eigenvalue
- that previous, important result in hand, we then dove into upper-triangular matricies
- specifically, we learned the properties of upper-triangular matrix, that if \(v_1 … v_{n}\) is a basis of \(V\) then \(\mathcal{M}(T)\) is upper-triangular if \(Tv_{j} \in span(v_1, … v_{j})\) for all \(j \leq n\); and, equivalently, \(T\) in invariant under the span of \(v_{j}\)
- using that result, we show that every complex operator has an upper-triangular matrix
- using some neat tricks of algebra, we then establish that operator is only invertible if diagonal of its upper-triangular matrix is nonzero, which seems awfully unmotivated until you learn that…
- eigenvalues of a map are the entries of the diagonal of its upper-triangular matrix, and that basically is a direct correlary from the upper-triangular matrix of \(T-\lambda I\)
New Definitions
- \(T^m\)
- \(p(T)\)
- technically also product of polynomials
- matrix of an operator
- diagonal of a matrix
- upper-triangular matrix
Results and Their Proofs
- \(p(z) \to p(T)\) is a linear function
- polynomial of operator is commutative
- operators on complex vector spaces have an eigenvalue
- properties of upper-triangular matrix
- every complex operator has an upper-triangular matrix
- operator is only invertible if diagonal of its upper-triangular matrix is nonzero
- eigenvalues of a map are the entries of the diagonal of its upper-triangular matrix
Questions for Jana
why define the matrix of an operator again??just to stress that its square- for the second flavor of the proof that every complex operator has an upper-triangular matrix, why is \(v_1 … v_{j}\) a basis of \(V\)?
Interesting Factoids
Its 12:18AM and I read this chapter for 5 hours. I also just got jumpscared by my phone notification. What’s happening?