User:Eml5526.s11.team4.simko/HW2

=Matrix Algebra=

Recall matrix algebra: (1) $$(\mathbf{A}\mathbf{B})^{T}=\boldsymbol{B}^{T}\boldsymbol{A}^{T}$$ (2) $$(\boldsymbol{A}^{-1})^{T}=(\boldsymbol{A}^{T})^{-1}=\boldsymbol{A}^{-T}$$ Given the matrices: $$ \boldsymbol{A}=\begin{bmatrix} 1 & 1 & 1\\ 2 & -1 & 3\\ 3 & 2 & 6 \end{bmatrix} $$ $$ \boldsymbol{B}=\begin{bmatrix} 1 & 3 & 5\\ 1 & -4 & 1\\ 2 & 5 & 8 \end{bmatrix} $$ {{1,1,1},{2,-1,3},{3,2,6}}*{{1,3,5},{1,-4,1},{2,5,8}}

http://www.wolframalpha.com/input/?i=matrix%20operations&lk=2 for matrix math

http://reference.wolfram.com/mathematica/ref/Transpose.html http://reference.wolfram.com/mathematica/tutorial/BasicMatrixOperations.html for transpose syntax

=Homework 2.2 (meeting 7)=

Problem Statement
Consider the family of vectors shown below in Equation 2.2-1.

These vectors exhibit properties such that

where $$\delta_{ij}$$ in Equation 2.2-2 is the Kronecker Delta defined as

The repeating index basis is given by Equation 2.2-4, where

The following two solution methods are observed in this problem set,

Complete the following tasks:


 * 1. Find $$ det[b_{jk}] $$


 * 2. Find $$ \mathbf{\Gamma(b_{1},b_{2},b_{3})}=\mathbf{K},\quad det(\mathbf{\Gamma}) $$


 * 3. Find $$ \mathbf{F}=\left\{F_{i}\right\}=\left\{\mathbf{b}_{i}\cdot \mathbf{v}\right\} $$


 * 4. Solve (2-4) for $$ \mathbf{d}=\left\{ v_{j} \right\} $$


 * 5. Use $$ \mathbf{w}_{i}\cdot \mathbf{\mathbb{P}(v)}=0 $$ to find (2-5). What is $$ \vec{\mathbf{K}} $$ and $$ \vec{\mathbf{F}}? \quad \mathbf{d}=\left\{ v_{j} \right\} $$


 * 6. Solve for $$ \mathbf{d} $$ ; Compare to $$ \mathbf{d} $$ in problem 4.


 * 7. Observe symmetry property of $$ \mathbf{K} $$ and $$ \vec{\mathbf{K}}. $$ Discuss pros and cons of 2 methods.(2-4 & 2-5)

1.) Find $$ det[b_{jk}] $$
In general the determinantof a 3X3 matrix is given by the following equation.

For the specific case of $$ [b_{jk}] $$ in Equation 2.2-4, the determinant is found to be equal to -8 ($$\textbf{Ans:} det[b_{jk}]=-8$$). This can also be calculated quickly using Matlab as shown below.  Matlab Code 

 Result 

2.) Find the Gram matrix $$ \mathbf{\Gamma(b_{1},b_{2},b_{3})}$$ and calculate the determinant.
From lecture slide 7-2,

The Gram matrix is given by $$ \mathbf{\Gamma(b_{1},b_{2},b_{3})}=\mathbf{K}_{ij}=\mathbf{[b_{i}]^{t}\cdot[b_{j}]}, \quad i,j=1,2,3 $$

where the stiffness matrix is given by,

with the vector components defined as


 * $$ \mathbf{b_{1}}=\begin{bmatrix}1 & 1 & 1\end{bmatrix}, \mathbf{b_{2}}=\begin{bmatrix}2 & -1 & 3\end{bmatrix}, \mathbf{b_{3}}=\begin{bmatrix}3 & 2 & 6\end{bmatrix} $$
 * }
 * }

Therefore,  Matlab Code 

 Result  And the determinant of $$ \mathbf{\Gamma} $$ is 64.