User:EGM6321.f12.team7.Marjanovic/report5/5.4

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Problem 4: Showing the Decomposition of a Matrix
Based on lecture notes section 20

Given: A Matrix
Also, according to section 20-4, a matrix can be decomposed as follows:

where the right-hand side of the equation consists of the eigenvector matrix, a diagonal eigenvalue matrix and the inverse of the eigenvector matrix, respectively.

Lastly,

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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To start off, we must find the eigenvalues of the matrix above

Taking the determinant and setting it equal to zero, we can see that we get

Using these eigenvalues, we get the following eigenvectors:

Next, using the method of Determinants to find the inverse of a matrix, which states for a matrix:

The inverse is found by:

Employing this technique on ($$) :

Lastly, our diagonal eigenvalue matrix is seen to be:

Therefore, the matrix can be written, in decomposed form, as:

which is identical to ($$)

Now for the second part, from ($$), we can see that the exponential of our matrix can be expressed as:

The diagonal matrix can be re-written using Euler's formula from (5) in section 27-6 to arrive at the final answer: