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#flo #hw
1 Eigen things!
instead of maps from vec space to another vec space, this is about a vec space to itself
the study of this is the most important part of linalg, according to axler
a subpsace which gets mapped into itself is an invariant subspace
ie, a subspace can be invarant under T if \(T|_U\) is an operator on \(U\)
pausing, pg 133
simplest possible nontrivial invariant subspace is an invariant subspace with dim 1
we can take any non-zero vector in V and make the set of all scalar multiples of it \(\text{span}(v) = U = \{ \lambda v : \lambda \in F \}\) thus, \(U\) must have dim 1, cus we only need the org vec to define the basis
if \(U\) is invariant under an operator, then that means that \(Tv \in U\) which means that there exists some scalar in F which does the same thing as the map \[Tv = \lambda v\]
pausing, again, pg. 134
we get the formal definiion of an eigenvalue, which is essentially the scalar from F that does the same thing as the transformation
we also get relations to surjectivity and injectivyt and invertibility and ect. KBrefSurjectiveFunction KBrefInjective KBrefInvertibleLinearMaps
an eigenvector is the corresponding vector that you pass through the linear map / \(\lambda\) ie, a vector that only get's scaled when you pass it through a linear map, as opposed to getting rotated or transalated
we also know that eigenvectors are LID when they have distinct eigenvalues and also, the number of distinc eigen values has to be \(\leq\) dim v
restirction operators and quotient operators restricted to U means you can only map into U? ie, instead of U -> V, it becomes U -> U?
1.1 5.B!
the reason why we care more about operators than normal linear maps is because operators can be raised to powers we can do \(T^{2}\), because we know it will be in \(\mathcal{L}(V)\) ! they follow all the normal exponentiation rules, except that \(T^0\) is the identity
and we get polynomial \(p(T)\) which is where instead of \(z \dots z^{m}\) we have T's
and also, operators on complex vector spaces have an eigenvalue
define diagonal of a matrix to be the diagonal of a sqaure matrix upper triangular is when all entries below the diag are 0. the upper-triangular matrix is usually represented in the form: \[ \begin{pmatrix} \lambda_1 & & * \\ & \ddots & \\ 0 & & \lambda_n \end{pmatrix}; \]
which is very pretty!
every operator over C has an upper triangular matrix we can also use these to determine where the operator is invertible. if the diag contains any 0's then it's non invertible!
relation to eigen values,
if the operator has an upper-triangular matrix w.r.t. some basis of V, then the eigenvalues of that operator are the entries on the diag.
and with that, we get to ## 5.C! diagonals and eigenspaces oh boy, here comes the real stuff!
diag matrix is a matrix with only non-zero numbers on the diag
the eigenvalues of an operator are the entries along the diag of a diag matrix
and we also get to define, eigenspace denoted \(E(\lambda, T)\) which is the set of all eigenvectors corresponding to \(\lambda\) + 0 vec (which is also a subspace)
and, ofc, the sum of the eigenspaces is a direct sum, as a single vec cannot be scaled by multiple \(\lambda\)'s
import def, diagonalizable operators on V can be diagonlizable if the op has a diag matrix w.r.t. some basis of V
also a bunch of things that are equivalent to being diagonalizable, shown in 5.41
1.2 5.C, inclass
#extract?
- linear algebra can be seen as an equivalence class between multiple
different forms
- algebra is about manipulating these different representations
- diagonal form is one of these forms. its one of the standard forms
1.2.1 so, what is diagonalization anyways?
how we actually compute the eigenvalues is not how axler computes the eigen values how we actully do it, is set \((A-\lambda I) v = 0\) this is also equaivalent to the inverse of the first term not existing and the determinant being 0
the determinant of a matrix is the product of its eigenvalues! whaaaat?
the charateristic equation of a matrix A becomes a polynomial of degree N
title: characteristic polynomial the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. -google
the trace is the sum of the diag