Andrew Thangaraj
Aug-Nov 2020
\(T:V\to V\), operator and \(U\subseteq V\), invariant subspace
Pick subspace \(W\) s.t. \(V=U\oplus W\)
Basis for \(V\): \(\{u_1,\ldots,u_k,w_1,\ldots,w_{n-k}\}\)
\(\{u_1,\ldots,u_k\}\): basis for \(U\), \(\{w_1,\ldots,w_{n-k}\}\): basis for \(W\)
Matrix of \(T\) in above basis
\(\begin{bmatrix} \vdots&\cdots&\vdots&\vdots&\cdots&\vdots\\ Tu_1&\cdots&Tu_k&Tw_1&\cdots&Tw_{n-k}\\ \vdots&\cdots&\vdots&\vdots&\cdots&\vdots \end{bmatrix}\)
What happens because of invariance of \(U\)?
\(Tu_i = a_{1i}u_1+\cdots+a_{ki}u_k+0w_1+\cdots+0w_{n-k}\)
Form of matrix for \(T\)
\(\begin{bmatrix} a_{11}&\cdots&a_{1k}&&\cdots&\\ \vdots&\vdots&\vdots&\vdots&\cdots&\vdots\\ a_{k1}&\cdots&a_{kk}&Tw_1&\cdots&Tw_{n-k}\\ 0&\cdots&0 &\vdots&\cdots&\vdots\\ \vdots&\vdots&\vdots&&\cdots&\\ 0&\cdots&0 &&\cdots& \end{bmatrix}\)
\(V\): over \(\mathbb{C}\). \(Tv_1=\lambda v_1\). Basis for \(V\): \(\{v_1,w_1,\ldots,w_{n-1}\}\)
\(Tw_j = b_{0j}v_1+b_{1j}w_1+\cdots+b_{n-1,j}w_{n-1}\)
\(\begin{bmatrix} \lambda&b_{0j}&\cdots&b_{0,n-1}\\ 0&b_{11}&\cdots&b_{1,n-1}\\ \vdots&\vdots&\cdots&\vdots\\ 0&b_{n-1,1}&\cdots&b_{n-1,n-1} \end{bmatrix}\)
\(\begin{bmatrix} b_{0j}&\cdots&b_{0,n-1} \end{bmatrix}\leftrightarrow T_0:W\to\) span\(\{v_1\}\) in basis \(\{w_1,\ldots,w_{n-1}\}\)
\(\begin{bmatrix} b_{11}&\cdots&b_{1,n-1}\\ \vdots&\cdots&\vdots\\ b_{n-1,1}&\cdots&b_{n-1,n-1} \end{bmatrix}\leftrightarrow T_1:W\to W\) in basis \(\{w_1,\ldots,w_{n-1}\}\)
\(v\in V\): \(v=c_1v_1+w'\), where \(w'=d_1w_1+\cdots+d_{n-1}w_{n-1}\in W\)
\(Tv=c_1\lambda v_1+T_0w'+T_1w'=(c_1\lambda+c_{w'})v_1+T_1w'\)
\(T_1w=\lambda_1 w\). Change basis for \(T\) from \(\{v_1,w_1,\ldots,w_{n-1}\}\) to \(\{v_1,w,w'_1,\ldots,w'_{n-2}\}\)
\(\begin{bmatrix} \lambda&*&*&\cdots&*\\ 0&\lambda_1&*&\cdots&*\\ 0&0&*&\cdots&*\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&*&\cdots&*\\ \end{bmatrix}\)
Every linear operator (over \(\mathbb{C}\)) has an upper triangular matrix representation.
Proof
Continue previous process - one eigenvalue at a time
All eigenvalues are on the diagonal in the upper triangular matrix representation
Geometric multiplicity of an eigenvalue \(\le\) algebraic multiplicity
Proof
\(A\): upper triangular matrix of \(T\)
AM\((\lambda)=\) number of times \(\lambda\) appears on diagonal
rank\((A-\lambda I)\ge n-\) AM\((\lambda)\)
Eigenspace of an eigenvalue \(\lambda\): \(E(\lambda, T)=\) null\((T-\lambda I)\)
Eigenspace is the set of all eigenvectors along with \(0\)
GM\((\lambda)=\) dim \(E(\lambda,T)\le\) AM\((\lambda)\)
dim \(E(\lambda,T)\): number of linearly independent eigenvectors
\(E(\lambda,T)\) and \(E(\lambda',T)\) intersect only at \(0\), if \(\lambda\ne \lambda'\)
\(T:V\to V\) with distinct eigenvalues \(\lambda_1,\ldots,\lambda_m\).
When is a linear map diagonalizable?
GM\((\lambda_i) =\) AM\((\lambda_i)\) for \(i=1,\ldots,m\)
\(V=E(\lambda_1,T)\oplus\cdots\oplus E(\lambda_m,T)\)