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HW4-1.pdf

EGN 4453 Homework 3

Due Friday, December 2rd, 2022 11:59pm

For all MATLAB codes written, appropriate comments are re- quired. Solutions must be uploaded in PDF format to Canvas.

1. Consider the following matrix

A =

[ 2 2 2 5

] (a) Find the characteristic polynomial.

(b) Find the eigenvalues.

(c) Compute the normalized eigenvector for each eigenvalue.

(d) Demonstrate your answers are correct by computing the Rayleigh Quotient.

2. Consider the following vectors:

a1 =

1 8 5

 ; a2 =

8 5 6

 ; a3 =

4 5 6

 î

Assume a reference (un-primed) coordinate system with unit basis vectors , ĵ, k̂. You will construct another coordinate system (primed) from the

vectors above with basis vectors ê1, ê2, ê3.

(a) Use the Gram-Schmidt procedure from Lab 7 to construct 3 mutually orthogonal vectors, beginning by making ê1 parallel to a1.

(b) Given the vector in the un-primed (̂i, ĵ, k̂) coordinate system

x =

 4 −3 2

 find its representation in the primed (ê1, ê2, ê3) coordinate system, i.e. find the values x′

1, x ′ 2, x

′ 3 so that

x′ = x′ 1ê1 + x′

2ê2 + x′ 3ê3

(c) Given the following vector in the primed (ê1, ê2, ê3) coordinate sys- tem

y′ =

1 2 3

 find its representation in the un-primed (̂i, ĵ, k̂) coordinate system, i.e. find the values y1, y2, y3 so that

y = y1 î + y2ĵ + y3k̂

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