Eigenvalues, Eigenvectors and Upper Triangularization

Aug-Nov 2020

Recap

• Vector space $V$ over a scalar field $F$
• $F$: real field $\mathbb{R}$ or complex field $\mathbb{C}$ in this course
• $m\times n$ matrix A represents a linear map $T:F^n\to F^m$
• dim null $T$ + dim range $T$ = dim $V$
• Linear equation: $Ax=b$
• Solution (if it exists): $u+$ null$(A)$
• Four fundamental subspaces of a matrix
• Column space, row space, null space, left null space
• Eigenvalue $\lambda$ and Eigenvector $v$: $Tv=\lambda v$
• Distinct eigenvalues have independent eigenvectors
• Basis of eigenvectors results in a diagonal matrix for $T$

Invariant subspaces and matrices of operators

$T:V\to V$, 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 $U$?

$Tu_i = a_{1i}u_1+\cdots+a_{ki}u_k+0w_1+\cdots+0w_{n-k}$

Invariant subspaces, eigenvalues and matrices of operators

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 $V$: $\{v_1,w_1,\ldots,w_{n-1}\}$

$Tw_j = b_{0j}v_1+b_{1j}w_1+\cdots+b_{n-1,j}w_{n-1}$

$\begin{bmatrix} \lambda&b_{0j}&\cdots&b_{0,n-1}\\ 0&b_{11}&\cdots&b_{1,n-1}\\ \vdots&\vdots&\cdots&\vdots\\ 0&b_{n-1,1}&\cdots&b_{n-1,n-1} \end{bmatrix}$

Towards upper triangularization

$\begin{bmatrix} b_{0j}&\cdots&b_{0,n-1} \end{bmatrix}\leftrightarrow T_0:W\to$ 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 $\{v_1,w,w'_1,\ldots,w'_{n-2}\}$

$\begin{bmatrix} \lambda&*&*&\cdots&*\\ 0&\lambda_1&*&\cdots&*\\ 0&0&*&\cdots&*\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&*&\cdots&*\\ \end{bmatrix}$

Upper triangularization

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

• Algebraic multiplicity of an eigenvalue: number of times it appears on the diagonal

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$(\lambda)$

Eigenspaces and diagonalization

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 $\lambda_1,\ldots,\lambda_m$.

• Eigenspaces: $E(\lambda_1,T), \ldots, E(\lambda_m,T)$

When is a linear map diagonalizable?

• GM$(\lambda_i) =$ AM$(\lambda_i)$ for $i=1,\ldots,m$

• $V=E(\lambda_1,T)\oplus\cdots\oplus E(\lambda_m,T)$