User:EML4500.f08.Zhichao Gong/CHAP 0

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CHAP 0 MATHEMATICAL PRELIMINARY FINITE ELEMENT ANALYSIS AND DESIGN Nam-Ho Kim Transcribed into mediawiki by Zhichao Gong

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Vector: a collection of scalars, defined using a bold typeface with braces

where the components $$\displaystyle a_1, a_2, a_3$$ had been arranged in matrix form as shown in Eq.($$).


 * Matrix- a collection of vectors, defined using a blod typeface with brackets - dimension = N*K, When N=K, it is called asquare matrix<P></P>

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- Transpose of a matrix: Change of row and column

- Symmetric and Skew-symmetric matrices

- Identity matrix

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<P align=center>VECTOR-MATRIX CALCULUS</P>

<UL> <LI>Addition</LI></UL>

<UL> <LI>Scalar product between two vectors (must be the same dim)</LI></UL>

<UL> <LI>Norm (Magnitude of a vector)</LI></UL>

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<UL> <LI>Similar to the norm of a vector</LI> <LI>Only defined for a square matrix</LI> <LI>If a determinant is zero, the matrix is not invertible</LI> <LI>A matrix is singular when its determinant is zero</LI> <LI>For 2*2 matrix</LI></UL>

<UL> <LI>For a 3*3 matrix</LI></UL>

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<P align=center><STRONG>VECTOR-MATRIX CALCULUS cont.</STRONG></P>

<UL> <LI>Vector product</LI></UL> <P> </P>

$$\displaystyle -\; Scalar \; product\rightarrow result\; =\; scalar $$

$$\displaystyle -\; Vector \; product\rightarrow result\; =\; vector $$

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<P align=center><STRONG>MATRIX-VECTOR MULTIPLICATION</STRONG></P>

$$\displaystyle \bullet Matrix\, \times \, Vector\, =\, Vector $$

{{NumBlk|:|$$\displaystyle c_{i}=\sum_{j=1}^{3}m_{ij}a_{j},\, \, i=1,2,3 $$|$$}

$$\displaystyle \bullet \, Vector\, \times \, Matrix \, \times \, Vector \, = \, Scalar $$

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<P align=center>MATRIX-MATRIX MULTIPLICATION</P>

$$\displaystyle \bullet \, Matrix \, \times \, Matrix \, =Matrix $$

<UL style="MARGIN-RIGHT: 0px" dir=ltr> <LI>Inverse of a matrix</LI></UL> <P dir=ltr>          - A square matrix [A] is invertible, then</P>

<P>           - If a matrix is singular ( |<STRONG>A</STRONG>| = 0 ), then the inverse does not exist</P>

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<P align=center><STRONG>RULES OF MATRIX MULTIPLICATION</STRONG></P>

<UL> <LI>Associative rule:</LI></UL>

$$\displaystyle \left ( AB \right )C=\left ( BC \right ) $$

<UL> <LI>Distributive rule:</LI></UL>

$$\displaystyle A\left ( B+C \right )=AB+AC $$

<UL> <LI>Non-commutative:</LI></UL>

$$\displaystyle AB\neq BA $$

<UL> <LI>Transpose of product:</LI></UL>

$$\displaystyle \left ( AB \right )^{T}=B^{T}A^{T}, \left ( ABC \right )^{T}=C^{T}B^{T}A^{T} $$

<UL> <LI>Inverse of product:</LI></UL>

$$\displaystyle \left ( AB \right )^{-1}=B^{-1}A^{-1}, \left ( ABC \right )^{-1}=C^{-1}B^{-1}A^{-1} $$

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<P align=center><STRONG>MATRIX EQUATION</STRONG></P>

- N unkowns $$\left ( x_{1},x_{2},\cdots ,x_{n} \right )$$ and N equations

- unique solution if all equations are independent

$$\begin{matrix} a_{11}x_{1}+a_{12}x_{2}+\cdots +a_{1N}x_{N}=b_{1}\\ a_{21}x_{1}+a_{22}x_{2}+\cdots +a_{2N}x_{N}=b_{2}\\ \vdots \\ a_{N1}x_{1}+a_{N2}x_{2}+\cdots +a_{NN}x_{N}=b_{N} \end{matrix} $$

$$\left [ \mathbf{A} \right ]=\begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1N}\\ a_{11} & a_{11} & \cdots & a_{11}\\ \vdots & \vdots & \ddots & \vdots\\ a_{N1} & a_{N2} & \cdots & a_{NN} \end{bmatrix}$$