array

(noun)

An ordered arrangement.

Related Terms

Examples of array in the following topics:

  • Examples

    • $\displaystyle{\left( \begin{array}{ccc} 1 & 0 & -1 \\ 0 & 1 & 2 \end{array} \right) \left( \begin{array}{c} x_1 \\ x_2 \\ x_3 \end{array} \right) = \left( \begin{array}{c} 0 \\ 0 \end{array} \right). }$
    • $\left\{ \left( \begin{array}{c} 1 \\ 1 \\ 0 \end{array} \right), \left( \begin{array}{c} 0 \\ 2 \\ 3 \end{array} \right), \left( \begin{array}{c} 1 \\ 2 \\ 3 \end{array} \right), \left( \begin{array}{c} 3\\ 6 \\ 6 \end{array} \right) \right\}$
    • $x \left( \begin{array}{c} 1 \\ 1 \\ 0 \end{array} \right) + y \left( \begin{array}{c} 0 \\ 2 \\ 3 \end{array} \right) + z \left( \begin{array}{c} 1 \\ 2 \\ 3 \end{array} \right) = \left( \begin{array}{c} 3\\ 6 \\ 6 \end{array} \right)$
    • $\left( \begin{array}{ccc} 1 & 0 & 1 \\ 1 & 2 & 2 \\ 0 & 3 & 3 \end{array} \right) \left( \begin{array}{c} x \\ y \\ z \end{array} \right) = \left( \begin{array}{c} 3 \\ 6 \\ 6 \end{array} \right)$
    • $2 \left( \begin{array}{c} 1 \\ 1 \\ 0 \end{array} \right) + 1 \left( \begin{array}{c} 0 \\ 2 \\ 3 \end{array} \right) + 1 \left( \begin{array}{c} 1 \\ 2 \\ 3 \end{array} \right) = \left( \begin{array}{c} 3\\ 6 \\ 6 \end{array} \right)$

  • Examples of Least Squares

    • $\left( \begin{array}{c} 1 \\ 1 \\ 1 \\ \vdots \\ 1 \end{array} \right) x = \left( \begin{array}{c} x_1 \\ x_2 \\ x_3 \\ \vdots \\ x_n \end{array} \right)$
    • $\left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \\ 0 & 2 \end{array} \right) \left( \begin{array}{c} x \\ y \end{array} \right) = \left( \begin{array}{c} \alpha \\ \beta \\ \gamma \end{array} \right)$
    • $\left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \\ 0 & 2 \end{array} \right) \left( \begin{array}{c} x \\ y \end{array} \right) = \left( \begin{array}{c} 0 \\ 0 \\ 1 \end{array} \right)$
    • $A^T A = \left( \begin{array}{cc} 1 & 1 \\ 1 & 6 \end{array} \right) .$
    • $\mathbf{x_{ls}} = \left( \begin{array}{ccc} 1 & -1/5 & -2/5 \\ 1 & 1/5 & 2/5 \end{array} \right) \left( \begin{array}{c} 0 \\ 0 \\ 1 \end{array} \right) = \left( \begin{array}{c} -2/5 \\ 2/5 \end{array} \right) .$

  • A few examples

    • $A = \left[ \begin{array}{cc} 3 & 1 \\ 0 & 3 \\ \end{array} \right]$
    • $\left[ \begin{array}{cc} 0 & 1 \\ 1 & 0 \\ \end{array} \right]?
    • $\left[ \begin{array}{ccc} -2 & 1& 0 \\ 1 &-2& 1 \\ 0 &1& -2 \end{array} \right] \left[ \begin{array}{c} a \\ b \\ c \end{array} \right] = \left[ \begin{array}{c} 0 \\ 0 \\ 0 \end{array} \right]$
    • $\left[ \begin{array}{cc} a & b\\ b & d \end{array} \right] \left[ \begin{array}{c} x \\ y \end{array} \right] = \left[ \begin{array}{c} 0 \\ 0 \end{array} \right]$
    • $\left[ \begin{array}{cc} 1 & -1 \\ 0 & 0 \end{array} \right] \mbox{ and } \left[ \begin{array}{ccc} 0 & 0 & 0\\ 0 & 0 & 0 \end{array} \right]$

  • A Matrix Appears

    • $\displaystyle{ \mathbf{u} = \left[ \begin{array}{c} A e ^{i \omega t} \\ B e ^{i \omega t} \end{array} \right] = e^ {i \omega t} \left[ \begin{array}{c} A \\ B \end{array} \right]. }$
    • $\displaystyle{ \mathbf{u} = e^{i \omega _ 0 t} \left[ \begin{array}{c} 1 \\ 1 \end{array} \right], }$
    • $\displaystyle{ \mathbf{u} = e^{i \sqrt{3} \omega _ 0 t} \left[ \begin{array}{c} 1 \\ -1 \end{array} \right]. }$
    • $\displaystyle{ \left[ \begin{array}{c} 1 \\ 1 \end{array} \right] \hbox{ and } \left[ \begin{array}{c} 1 \\ -1 \end{array} \right] }$
    • $\left( \left[ \begin{array}{c} 1 \\ 1 \end{array} \right] \cdot \left[ \begin{array}{c} 1 \\ -1 \end{array} \right] \equiv \left[ 1,1 \right] \left[ \begin{array}{c} 1 \\ -1 \end{array} \right] = 1\cdot 1 - 1 \cdot 1 = 0.

  • Matrices

    • $A= \left[ \begin{array}{cc} 2 & 5 \\ 3 & 8 \\ 1 & 0 \end{array} \right].$
    • $A^T = \left[ \begin{array}{ccc} 2 & 3 & 1\\ 5 & 8 & 0 \end{array} \right].$
    • $A \cdot \mathbf{x} = \left[ \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \\ a_{31} & a_{32} \end{array} \right] \cdot \left[ \begin{array}{c} x_1 \\ x_2 \end{array} \right] = x_1 \left[ \begin{array}{c} a_{11} \\ a_{21} \\ a_{31} \end{array} \right] + x_2 \left[ \begin{array}{c} a_{12} \\ a_{22} \\ a_{32} \end{array} \right]$
    • $AB = \left[ \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array} \right] \left[ \begin{array}{cc} 0 & 1 \\ 2 & 3 \end{array} \right] = \left[ \begin{array}{cc} 4 & 7 \\ 8 & 15 \end{array} \right] .$
    • $BA = \left[ \begin{array}{cc} 0 & 1 \\ 2 & 3 \end{array} \right] \left[ \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array} \right] = \left[ \begin{array}{cc} 3 & 4 \\ 11 & 16 \end{array} \right] \neq AB.$

  • Matrices for two degrees of freedom

    • $\displaystyle{ \left[ \begin{array}{cc} m_1 & 0 \\ 0 & m_2 \end{array} \right] \left[ \begin{array}{c} \ddot{x}_1 \\ \ddot{x}_2 \end{array} \right] + \left[ \begin{array}{cc} k_1 + k_2 & -k_2 \\ -k_2 & k_2+k_3 \\ \end{array} \right] \left[ \begin{array}{c} x_1 \\ x_2 \end{array} \right] = \left[ \begin{array}{c} 0 \\ 0 \end{array} \right] . }$
    • $\displaystyle{ M = \left[ \begin{array}{cc} m_1 & 0 \\ 0 & m_2 \end{array} \right] }$
    • $\displaystyle{ K = \left[ \begin{array}{cc} k_1 + k_2 & -k_2 \\ -k_2 & k_2+k_3 \\ \end{array} \right] }$
    • $\displaystyle{ \mathbf{u} \equiv \left[ \begin{array}{c} x_1 \\ x_2 \end{array} \right]. }$
    • $\displaystyle{ M^{-1} K = \Omega ^2 \left[ \begin{array}{cc} 1 & -1 \\ -1 & 1 \end{array} \right] + \omega_0 ^2 \left[ \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right]. }$

  • Eigenvalues and Eigenvectors

    • $\left( \begin{array}{cc} 2 \omega_0 ^2 & -\omega_0 ^2 \\ -\omega_0 ^2 & 2 \omega_0 ^2 \end{array} \right) \left( \begin{array}{c} A \\ B \end{array} \right) = \omega ^2 \left( \begin{array}{c} A \\ B \end{array} \right).$
    • $ \left( \begin{array}{cc} 5 & 1 \\ 1 & 5 \end{array} \right) \left( \begin{array}{c} y \\ y \end{array} \right) = 6 \left( \begin{array}{c} y \\ y \end{array} \right)$
    • $\left( \begin{array}{cc} 5 & 1 \\ 1 & 5 \end{array} \right) \left( \begin{array}{c} y \\ y \end{array} \right) = 4 \left( \begin{array}{c} y \\ y \end{array} \right)$
    • $\left[ \begin{array}{cc} 2\sqrt{3} & 3 \\ 4 & 2\sqrt{3} \end{array} \right] \left[ \begin{array}{c} x_1 \\ x_2 \end{array} \right] =0$
    • $\left[ \begin{array}{cc} -2\sqrt{3} & 3 \\ 4 & -2\sqrt{3} \end{array} \right] \left[ \begin{array}{c} x_1 \\ x_2 \end{array} \right] =0$

  • The Cross Product

    • $=\left[ {\begin{array}{cc} a_2 & a_3 \\ b_2 & b_3 \\ \end{array} } \right]i - \left[ {\begin{array}{cc} a_1 & a_3 \\ b_1 & b_3 \\ \end{array} } \right]j + \left[ {\begin{array}{cc} a_1 & a_2 \\ b_1 & b_2 \\ \end{array} } \right]k\\ =\left[ {\begin{array}{cc} i & j & k \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ \end{array} } \right]\\ $

  • B.4 Chapter 4

    • $\displaystyle U_{l}^\mu = \left [ \begin{array}{c} c \\ 0 \\ 0 \\ 0 \end{array} \right ]$
    • $\displaystyle p^\mu = \left [ \begin{array}{c} m\gamma c \\ -m\gamma v \\ 0 \\ 0 \end{array} \right ]$
    • $\displaystyle p^\mu = \left [ \begin{array}{c} m\gamma c \\ m\gamma v \\ 0 \\ 0 \end{array} \right ]$
    • $\displaystyle U^\mu_r p_\mu = \left [ \begin{array}{c} \gamma c \\ -\gamma v \\ 0 \\ 0 \end{array} \right ] \left [ \begin{array}{cccc} m\gamma c & -m\gamma v & 0 & 0 \end{array} \right ] = m \gamma^2 \left ( c^2 + v^2 \right )$
    • $\displaystyle p^\mu = \frac{E}{c} \left [ \begin{array}{c} 1 \\ {\bf n} \end{array} \right ] ~\mathrm{Take}~ p^\mu = \left [ \begin{array}{c} \frac{E}{c} \\ \frac{E}{c} \\ 0 \\ 0 \\ \end{array} \right ]$

  • Tensors

    • $\displaystyle A^\mu = \left [ \begin{array}{c} \phi \\ {\bf A} \end{array} \right ],$
    • $\displaystyle J^\mu = \left [ \begin{array}{c} c \rho \\ {\bf J} \end{array} \right ].$
    • $\displaystyle F_{\alpha\beta} = -\left ( A_{\alpha,\beta} - A_{\beta,\alpha} \right ) = \left [ \begin{array}{cccc} 0 & -E_x & -E_y & -E_z \\ E_x & 0 & B_z & -B_y \\ E_y & -B_z & 0 & B_x \\ E_z & B_y & -B_x & 0 \end{array} \right ]$
    • $\displaystyle \frac{d}{d\tau} \left [ \begin{array}{c} E \\ p_x \\ p_y \\ p_z \end{array} \right ] = \frac{\gamma q}{c} \left [ \begin{array}{cccc} 0 & E_x & E_y & E_z \\ E_x & 0 & B_z & -B_y \\ E_y & -B_z & 0 & B_x \\ E_z & B_y & -B_x & 0 \end{array} \right ] \left [ \begin{array}{c} c \\ v_x \\ v_y \\ v_z \end{array} \right ]$
    • $\displaystyle {\cal F}^{\alpha\beta} = \frac{1}{2} \epsilon^{\alpha\beta\gamma\delta}F_{\gamma\delta} = \left [ \begin{array}{cccc} 0 & -B_x & -B_y & -B_z \\ B_x & 0 & E_z & -E_y \\ B_y & -E_z & 0 & E_x \\ B_z & E_y & -E_x & 0 \end{array} \right ]$