Linear Algebra Basics

Matrix-Vector multiplication

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\begin{equation} \left\{\begin{matrix} c_1 \\ c_2 \\ c_3 \end{matrix}\right\} = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} \left\{\begin{matrix} b_1 \\ b_2 \\ b_3 \end{matrix}\right\} \end{equation}

$$ c_1 = a_{11} b_1 + a_{12} b_2 + a_{13} b_3 $$

$$ c_2 = a_{21} b_1 + a_{22} b_2 + a_{23} b_3 $$

$$ c_3 = a_{31} b_1 + a_{32} b_2 + a_{33} b_3 $$

In words: $c_i$ is the dot product of the $i^{\mbox{th}}$ row of $\mathbf{a}$ with $\vec{b}$...

Matrix-Matrix multiplication

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\begin{equation} \begin{bmatrix} c_{11} & c_{12} & c_{13} \\ c_{21} & c_{22} & c_{23} \\ c_{31} & c_{32} & c_{33} \end{bmatrix} = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} \begin{bmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} & b_{32} & b_{33} \end{bmatrix} \end{equation}

$$ c_{11} = a_{11} b_{11} + a_{12} b_{21} + a_{13} b_{31} $$

\begin{align} c_{12} &= a_{11} b_{12} + a_{12} b_{22} + a_{13} b_{32} \\ & \vdots \end{align}

In words: $c_{ij}$ is the dot product of the $i^{\mbox{th}}$ row of $\mathbf{a}$ with the $j^{\mbox{th}}$ column of $\mathbf{b}$

Examples

\begin{equation} \begin{bmatrix} 2 & 1 \\ 3 & 2 \end{bmatrix} \left\{ \begin{matrix} 1 \\ 2 \end{matrix}\right\} = \left\{\begin{matrix} 4 \\ 7 \end{matrix}\right\} \end{equation}

\begin{equation} \begin{bmatrix} 2 & 1 \\ 3 & 2 \end{bmatrix} \begin{bmatrix} 1 & 3 \\ 1 & 2 \end{bmatrix} = \begin{bmatrix} 3 & 8 \\ 5 & 13 \end{bmatrix} \end{equation}

The determinant of a 2 x 2 matrix

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$$ \mathbf{A} = \begin{bmatrix} a & b \\ c & d \end{bmatrix} $$

$$ \det(\mathbf{A}) = a \cdot d - b \cdot c $$

The determinant of a 3 x 3 matrix

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$$ \mathbf{A} = \begin{bmatrix} a & b & c\\ d & e & f\\ g & h & i \end{bmatrix} $$

$$ \det(\mathbf{A}) = a \cdot ( e \cdot i - f \cdot h ) - b \cdot ( d \cdot i - f \cdot g ) + c \cdot ( d \cdot h - e \cdot g) $$

Matrix row operations

Used in solving matrix equations, i.e.

\begin{equation} \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} \left\{ \begin{matrix} x_1 \\ x_2 \\ x_3 \end{matrix} \right\} = \left\{\begin{matrix} b_1 \\ b_2 \\ b_3 \end{matrix} \right\} \end{equation}

$$\mathbf{A}\vec{x} = \vec{b}$$

• Swaping rows doesn't change solution

• Adding rows together doesn't change solution

• Multiplying row by a scalar doesn't change solution

Example

Solve for $\vec{x}$

\begin{equation} \begin{bmatrix} 2 & 3 & -2 \\ 0 & 0 & 3 \\ 1 & 0 & 2 \\ \end{bmatrix} \left\{ \begin{matrix} x_1 \\ x_2 \\ x_3 \end{matrix}\right\} = \left\{ \begin{matrix} 6 \\ -6 \\ 3 \end{matrix}\right\} \end{equation}

Eigenvalue problem

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$$ \mathbf{A} \vec{v} = \lambda \vec{v} $$

A vector $\vec{v}$ is said to be an eigenvector of $\mathbf{A}$ if when multiplied by $\mathbf{A}$ on the right, it becomes a scalar multple of itself.

$$ \mathbf{A}\vec{v} - \lambda \vec{v} = 0 $$

$$ (\mathbf{A} - \lambda \mathbf{I})\vec{v} = 0 $$

A non-trivial solution for $\vec{v}$ exists, if and only if

$$ \det(\mathbf{A} - \lambda \mathbf{I}) = 0 $$

The $\lambda$'s are called the eigenvalues. Examples to follow in the context of stress.

Vector transformation

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$$ \vec{v}' = \mathbf{Q} \vec{v} $$

$$ \mathbf{Q} = \begin{bmatrix} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & \cos(\theta) \end{bmatrix} $$

Matrix transformation

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$$ \mathbf{S}' = \mathbf{Q}^{-1} \mathbf{S} \mathbf{Q} $$

If $\mathbf{Q}$ is chosen such that its columns are eigenvectors of $\mathbf{S}$, then $\mathbf{S}'$ will be diagonal with its entries cooresponding to the eigenvalues $\mathbf{S}$ (and $\mathbf{S}'$).