120 lines
3.1 KiB
Markdown
120 lines
3.1 KiB
Markdown
# Lecture 38
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## Chapter VIII Operators on complex vector spaces
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### Trace 8D
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#### Definition 8.47
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For a square matrix $A$, the **trace of** $A$ is the sum of the diagonal entries denoted $tr(A)$.
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#### Theorem 8.49
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Suppose $A$ is $m\times n$, $B$ is $n\times m$ matrices, then $tr(AB)=tr(BA)$.
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Proof:
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By pure computation.
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#### Theorem 8.50
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Suppose $T\in \mathscr{L}(V)$ and $u_1,...,u_n$ and $v_1,...,v_n$ are bases of $V$.
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$$
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tr(M(T,(u_1,...,u_n)))=tr(M(T,(v_1,...,v_n)))
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$$
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Proof:
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Let $A=tr(M(T,(u_1,...,u_n)))$ and $B=tr(M(T,(v_1,...,v_n)))$, then there exists $C$, invertible such that $A=CBC^{-1}$,
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$$
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tr(A)=tr((CB)C^{-1})=tr(C^{-1}(CB))=tr(B)
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$$
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#### Definition 8.51
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Given $T\in \mathscr{L}(V)$ the trace of $T$ denoted $tr(T)$ is given by $tr(T)=tr(M(T))$.
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Note: For an upper triangular matrix, the diagonal entries are the eigenvalues with multiplicity
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#### Theorem 8.52
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Suppose $V$ is a complex vector space such that $T\in \mathscr{L}(V)$, then $tr(T)$ is the sum of the eigenvalues counted with multiplicity.
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Proof:
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Over $\mathbb{C}$, there is a basis where $M(T)$ is upper triangular.
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#### Theorem 8.54
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Suppose $V$ is a complex vector space, $n=dim\ V$.$T\in \mathscr{L}(V)$. Then the coefficient on $z^{n-1}$ in the characteristic polynomial is $tr(T)$.
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Proof:
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$(z-\lambda_1)\dots(z-\lambda_n)=z^{n}-(\lambda_1+...+\lambda_n)z^{n-1}+\dots$
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#### Theorem 8.56
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Trance is linear
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Proof:
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- Additivity
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$tr(T+S)=tr(M(T)+M(S))=tr(T)+tr(S)$
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- Homogeneity
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$tr(cT)=ctr(M(T))=ctr(T)$
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#### Theorem/Example 8.10
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Trace is the unique linear functional $\mathscr{L}\to \mathbb{F}$ such that $tr(ST)=tr(TS)$ and $tr(I)=dim\ V$
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Proof:
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Let $\varphi:\mathscr{L}(V)\to \mathbb{F}$ be a linear functional such that $\varphi(ST)=\varphi(TS)$ and $\varphi(I)=n$ where $n=dim\ V$. Let $v_1,...,v_n$ be a basis for $V$ define $P_{j,k}$ to be the operator $M(P_{j,k})=\begin{pmatrix}
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0&0&0\\
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0&1&0\\
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0&0&0
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\end{pmatrix}$. Note $P_{j,k}$ form a basis of $L(V)$, now we must show $\varphi(P_{j,k})=tr(P_{j,k})=\begin{cases}1\textup{ if }j=k\\0\textup{ if }j \neq k\end{cases}$
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- For $j\neq k$
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$\varphi(P_{j,j}P_{j,k})=\varphi(P_{j,k})=0$
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$\varphi(P_{j,k}P_{j,j})=\varphi(P_{j,k})=0$
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- For $j=k$
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$\varphi(P_{k,j},P_{j,k})=\varphi(P_{k,k})=1$
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$\varphi(P_{j,k},P_{k,j})=\varphi(P_{j,j})=1$
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So $\varphi(I)=\varphi(P_{1,1}+...+P_{n,n})=\varphi(P_{1,1})+...+\varphi(P_{n,n})=n$
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#### Theorem 8.57
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Suppose $V$ is finite dimensional vector space, then there does not exists $S,T\in \mathscr{L}(V)$ such that $ST-TS=I$. ($ST-TS$ is called communicator)
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Proof:
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$tr(ST-TS)=tr(ST)-tr(TS)=tr(ST)-tr(ST)=0$, since $tr(I)=dim\ V$, so $ST-TS\neq I$
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Note: **requires finite dimensional.**
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## Chapter ? Multilinear Algebra and Determinants
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### Determinants ?A
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#### Definition ?.1
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The determinant of $T\in \mathscr{L}(V)$ is the product of eigenvalues counted with multiplicity.
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#### Definition ?.2
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The determinant of a matrix is given by
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$$
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det(A)=\sum_{\sigma\in perm(n)}A_{\sigma(1),1}\cdot ...\cdot A_{\sigma(n),n}\cdot sign(\sigma)
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$$
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$perm(\sigma)=$ all recordings of $1,...,n$, number of swaps needed to write $\sigma$
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$$
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