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{\Huge 189-251B: Algebra 2} \\ \vskip 0.1in
{\Huge Assignment 7} \\ \vskip 0.1in
{\Large Due: Wednesday, February 26}
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1. Let $T:V\lra V$ be a linear transformation of finite-dimensional vector spaces
over a field $F$ which is not necessarily algebraically closed, and let
$p(x) = p_1(x) \cdots p_r(x)$ be the factorisation of $p(x)$ into monic irreducible factors.
a) Assume that the factors $p_j(x)$ are all distinct. Show that there is a direct sum decomposition
$$ V = V_1 \oplus \cdots \oplus V_r $$
into $F$-vector subspaces $V_j$ which are stable under $T$, and such that
the restriction $T_j$ of $T$ to $V_j$ has $p_j(x)$ as its minimal polynomial.
b) If $\dim_F(V_j) = \deg(p_j)$, show that $V_j$ cannot be further decomposed into a direct sum of $T$-stable
$F$-vector subspaces.
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2. Let $f(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_k x^k + \cdots $ be a power series
in $x$ with complex coefficients and suppose that $\lambda\in \C$ belongs to the disk of absolute convergence for $f$.
Let $M(\lambda,d)$ be the $d\times d$ Jordan matrix with eigenvalue $\lambda$:
$$ M(\lambda,d) = \left(\begin{array}{cccccc}
\lambda & 1 & 0 & 0 & \cdots & 0 \\
0 & \lambda & 1 & 0 & \cdots & 0 \\
0 & 0 & \lambda & 1 & \cdots & 0 \\
\vdots & \vdots & \vdots & \vdots & & \vdots \\
0 & 0 & 0 & 0 & \cdots & \lambda
\end{array}\right).$$
Show that the infinite sum
$$f(M(\lambda,d)) = a_0 + a_1 M(\lambda,d) + a_2 M(\lambda,d)^2 + \cdots $$
converges to a $d\times d$ matrix, and give a closed form expression for this matrix. (You may find it helpful to consider
first the simplest cases where
$d =2 $ and $d=3$ in order to guess the general pattern.)
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3. Let $e^x$ denote the
exponential function defined by the power series
$$e^x = 1 + x + \frac{x^2}{2!} + \cdots + \frac{x^k}{k!} + \cdots $$
Using exercise 2 and the theory of the Jordan canonical form, show that the infinite sum defining $e^M$ converges, for any
$d\times d$ matrix $M$ with complex entries, and describe an algorithm to compute it.
Use your algorithm to compute the exponential of the $2\times 2$ matrix $\left(\begin{array}{cc} 1 & -1 \\ 1 & 3 \end{array}\right)$.
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