User:EGM6321.f12.team7.Zhou/R5

Problem 3: Proof property of eigenvalue matrix
Report problem 5.3 from section 20-6

Given:The expression of eigenvalue matrix, the diagonal matrix and exponentiation of matrix
The exponentiation of matrix is given by,

For a matrix$$A$$,

The eigenvalue matrix is made of matrix of eigenvector as shown below.Suppose we have a matrix$\mathbf{A}$. The eignvector is given by,

$\phi$ is eigenvector, If there are n linearly independent eigenvectors, we will have,

So the the equation{}can be written as,

Where,

The matrix can be diagonalizable, which is given by,

Find: The expression of exponentiation of matrix can also be diagonalizable
$$\exp \mathbf A =\mathbf \phi \, \text{Diag}[e^{\lambda_1}\cdots e^{\lambda_n}]\, \mathbf \phi ^{-1}$$

Solution

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On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions. Use equation ($$) into equati($$), we have,
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Since $$\mathbf \phi \, \mathbf \phi^{-1} =1$$, we have,

According to associative property, which is given by,

where $$\mathbf A, \mathbf B ,\mathbf C, \mathbf B \in \mathbf E^{n\times n}$$. So the equation turns into,

In the problem 2, we have already prove that,

So the equation turns into,

Problem 9: Use Maclaurin series to expand equations
Report problem 5.9 from section 64

Find: Maclaurin series expansion and definition of hypergeometric function
The Maclaurin series is an Taylor series expansion of function at x about 0. Which is given by,

Where, $$f^n(x) = \frac{d^n f(x)}{dx^n} $$

Two functions are given by,

$$(1-x)^{-a}$$

$$\frac{1}{x}\arctan (1+x)$$

The notation of hypergeometric function is given by,

Where $$-1< x < 1$$,

$$\begin{cases} a(0):=1\\a(k)=a(a+1)\cdots(a+k-1)\end{cases}$$

Solution

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On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.
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First function
SInce $$1-x=1$$. the expansion of this function is given by,

According to the notation,when $$ b = c=d$$, the equation satisfied, so the expression is given by,

Second function
This problem seems like incorrect. The zero point is not in the definition domain. If you have solution, fill in the blank right here.