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12.2: Matrix Multiplication Applications

  • Page ID
    88004
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    By Carey Smith

    Rotation Matrices

    One use of matrices is to rotate vectors.

    A 2D Rotation matrix is of this form:

    Mrot2D = [ cosd() -sind() ]

    [ sind() cosd() ]

    (Use cosd() and sind() when the angle is in degrees.)

    Use cos() and sin() when the angle is in radians.)

    Example \(\PageIndex{3}\) 2D Rotate by 60 degrees clockwise:

    Mrot2D60 = [ cosd(60) -sind(60) ] = [ 0.5 -0.866 ]

    [ sind(60) cosd(60) ] [ 0.866 0.5 ]

    Example code is in the attached file Ex_Rotation2D.m

    Solution

    Add example text here.

    3D Rotation matrices

    A 3D Rotation matrix about the z-axis is has a 2D rotation sub matrix in it. It is of this form::

    Mrot3Dz = [ cosd(z) -sind(z) 0 ]

    [ sind(z) cosd(z) 0 ]

    [ 0 0 1 ]

    3D Rotation matrix about the X-axis is:

    Mrot3Dx = [ 1 0 0 ]

    [ 0 cosd(x) -sind(x) ]

    [ 0 sind(x) cosd(x) ]

    3D Rotation matrix about the Y-axis is:

    Mrot3Dy = [ cosd(y) 0 sind(y) ]

    [ 0 1 0 ]

    [ -sind(y) 0 cosd(y) ]

    Example code is in the attached file Ex_Rotation3D.m

    This produced the following figures:

    Initial3D_vector.png

    Rotate3Dy.png

    Rotate3Dz.png

    Rotate3Dx.png

    Initial vector is on the Z-axis

    Rotate by+90 about Y it goes to the X-axis

    Rotate by+90 about Z it goes to the Y-axis

    Rotate by+90 about X it goes to the Z-axis

    Matrix Powers

    A matrix can be raised to a power using matrix multiplication.

    Example \(\PageIndex{4}\)

    A = [1 2
    3 4]

    A^2 = A*A = [1 2]*[1 2] = [ 7 10]

    [3 4] [3 4] [15 22]

    A^4 = [199 290]
    [435 634]

    Solution

    Add example text here.

    Life Cycle Matrix

    A Matrix can be used to model the number of plants or animals at various stages from year to year.

    Example \(\PageIndex{5}\) Draba Plant Matrix Life Cycle

    This example considers a small flowering plant called Lake Tahoe Draba. Information about this plant and an image is available at: https://calscape.org/Draba-asterophora-(). That image is copyrighted. The image below is a creative commons image of a similar draba species. The numbers used in this example are made up simply to illustrate a possible application of matrix multiplication.

    Draba_jorullensis.jpg

    https://commons.wikimedia.org/wiki/F...orullensis.jpg, Tim Park

    This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.

    Consider these stages of a plant:

    • Seed
    • Seedling (1st year)
    • Mature plant

    The numbers of plants in each stage will be the elements of a vector, D, for Draba.

    Initialize D to have 20 mature plants, with zeros for the other stages:

    D1 = [0

          0

          20]

    Let T = the transition matrix from 1 year to the next.

    Assume the following average numbers: (These are made up)1 plant produces 8 seeds.

    Fraction of seeds that become seedlings = 0.2

    Fraction of seedlings that become mature plants = 0.5

    Fraction of plants that survive from year to year = 0.7

    Create the transition matrix's rows:

    seeds = 8*number of mature plants:

    -> T_seeds = [0 0 8]

    seedlings = 0.2*number of mature seeds

    -> T_seedlings = [0.2 0 0]

    New mature plants from seedling = 0.5*seedlings

    0.7*mature plants survive from the previous year

    T_mature = [0 0.5 0.7]

    The T matrix is created by combining these 3 vectors:

    T = [T_seeds

    T_seedlings

    T_mature ]

    T= [0   0   8

        0.2 0   0

        0   0.5 0.7 ]

    Compute the numbers of each stage in year 2:

    D2 = T*D1

    Compute the numbers of each stage in year 3:

    D3 = T*D2

    Compute the numbers of each stage in year 4:

    D4 = T*D3

    Compare D4 another way using a matrix power:

    D4b = (T^3)*D1

    Verify that this produces the same result.

    Compute the numbers of each stage in year 10:

    D10 = (T^9)*D1

    Solution

    Add example text here.

    .


    This page titled 12.2: Matrix Multiplication Applications is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Carey Smith.