126 lines
3.6 KiB
Markdown
126 lines
3.6 KiB
Markdown
# Lecture 37
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## Chapter VIII Operators on complex vector spaces
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### Generalized Eigenspace Decomposition 8B
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---
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Review
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#### Definition 8.19
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The generalized eigenspace of $T$ for $\lambda \in \mathbb{F}$ is $G(\lambda,T)=\{v\in V\vert (T-\lambda I)^k v=0\textup{ for some k>0}\}$
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#### Theorem 8.20
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$G(\lambda, T)=null((T-\lambda I)^{dim\ V})$
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---
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New materials
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#### Theorem 8.31
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Suppose $v_1,...,v_n$ is a basis where $M(T,(v_1,...,v_k))$ is upper triangular. Then the number of times $\lambda$ appears on the diagonal is the multiplicity of $\lambda$ as an eigenvalue of $T$.
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Proof:
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Let $\lambda_1,...,\lambda_n$ be the diagonal entries, $S$ be such that $M(S,(v_1,...,v_n))$ is upper triangular. Note that if $\mu_1,...,\mu_n$ are the diagonal entires of $M(S)$, then the diagonal entires of $M(S^n)$ are $\mu_1^n,...,\mu_n^n$
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$$
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\begin{aligned}
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dim(null\ S^n)&=n-dim\ range\ (S^n)\leq n-\textup{ number of non-zero diagonal entries on } S^n\\
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&=\textup{ number of zero diagonal entries of }S^n
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\end{aligned}
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$$
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plus in $S=T-\lambda I$, then
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$$
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\begin{aligned}
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dim G(\lambda, T)&=dim(null\ (T-\lambda I)^n)\\
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&\leq \textup{number times where }\lambda \textup{ appears on the diagonal of }M(T)\\
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\end{aligned}
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$$
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Note:
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$V=G(\lambda_1, T)\oplus \dots \oplus G(\lambda_k, T)$
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for distinct $\lambda_1,...,\lambda_k$ thus $n=dim\ G(\lambda_1,T)+\dots +dim\ (\lambda_k, T)$
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on the other hand $n=\textup{ number of times }\lambda_1 \textup{ appears as a diagonal entry}+\dots +\textup{ number of times }\lambda_k \textup{ appears as a diagonal entry}+\dots $
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So $dim\ G(\lambda_i, T)=$ number of times where $\lambda_i$ appears oas a diagonal entry.
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#### Definition 8.35
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A **block diagonal matrix** is a matrix of the form $\begin{pmatrix}
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A_1& & 0\\
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& \ddots &\\
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0& & A_m
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\end{pmatrix}$ where $A_k$ is a **square matrix**.
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Example:
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$
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\begin{pmatrix}
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1&0&0 & 0&0\\
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0 & 2 &1&0&0\\
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0 & 0 &2&0&0\\
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0& 0&0& 4&1\\
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0& 0&0& 0&4\\
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\end{pmatrix}$
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#### Theorem
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Let $V$ be a complex vector space and let $\lambda_1,...,\lambda_m$ be the distinct eigenvalue of $T$ with multiplicity $d_1,...,d_m$, then there exists a basis where $\begin{pmatrix}
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A_1& & 0\\
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& \ddots &\\
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0& & A_m
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\end{pmatrix}$ where $A_k$ is a $d_k\times d_k$ matrix upper triangular with only $\lambda_k$ on the diagonal.
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Proof:
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Note that $(T-\lambda_k I)\vert_{G(\lambda_k,T)}$ is nilpotent. So there is a basis of $G(\lambda_k,T)$ where $(T-\lambda_k I)\vert_{G(\lambda_k,T)}$ is upper triangular with zeros on the diagonal. Then $(T-\lambda_k I)\vert_{G(\lambda_k,T)}$ is upper triangular with $\lambda_k$ on the diagonal.
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### Jordan Normal Form 8C
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Nilpotent operators
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Example: $T(x,y,z)=(0,x,y), M(T)=\begin{pmatrix}
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0&1&0\\
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0&0&1\\
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0&0&0
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\end{pmatrix}$
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#### Definition 8.44
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Let $T\in \mathscr{L}(V)$ a basis of $V$ is a **Jordan basis** of $T$ if in that basis $\begin{pmatrix}
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A_1& & 0\\
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& \ddots &\\
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0& & A_p
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\end{pmatrix}$ where each $A_k=\begin{pmatrix}
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\lambda_1& 1& & 0\\
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& \ddots& \ddots &\\
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&&\ddots& 1\\
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0&&&\lambda_k\\
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\end{pmatrix}$
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#### Theorem 8.45
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Suppose $T\in \mathscr{L}(V)$ is nilpotent, then there exists a basis of $V$ that is a Jordan basis of $T$.
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Sketch of Proof:
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Induct on $dim\ V$, if $dim\ V=1$, clear.
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if $dim\ V>1$, then let $m$ be such that $T^m=0$ and $T^{m-1}\neq 0$. Then $\exists u\in V$ such that $T^{m-1}u\neq 0$, then $Span (u,Tu, ...,T^{m-1}u)$ is $m$ dimensional.
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#### Theorem 8.46
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Suppose $V$ is a complex vector space $T\in \mathscr{L}(V)$ then $T$ has a Jordan basis.
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Proof:
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take $V=G(\lambda_1, T)\oplus \dots \oplus G(\lambda_m, T)$, then look at $(T-\lambda_k I)\vert_{G(\lambda_k,T)}$ |