This will make more sense with the visuals in the following sections. Transforming $v$ by multiplying it by the transformation matrix $A$ or its associated eigenvalue $\lambda$ will result in the same vector. The basic equation is:Īny vector $v$ on the line made from the points passing through the origin $(0,0)$ and an eigenvector are all eigenvectors. $x$ is called an eigenvector that when multiplied with $A$, yields a scalar value, $\lambda$, called the eigenvalue. So, the transformed matrix can be represented by the equation: Quick recap, a non-zero matrix $x$ can be transformed by multiplying it with a $n \times n$ square matrix, $A$. Here, we build on top of that and understand eigenvectors and eigenvalues visually. Previously, I wrote about visualising matrices and affine transformations. Plots(,Pyy],i=1.Eigenvectors and eigenvalues are used in many engineering problems and have applications in object recognition, edge detection in diffusion MRI images, moments of inertia in motor calculations, bridge modelling, Google’s PageRank algorithm and more on wikipedia. Plots(,data],i=1.tmax)],symbol=DIAMOND,symbolsize=7) Plots(,Pyy],i=1.512)],style=line) Īlternatively, the same calculations can be done by using MATLAB® syntax almost entirely.ĮvalM("x=sin(2*pi*50*t) + sin(2*pi*120*t)") Plots(,data],i=1.nops(t))],symbol=DIAMOND,symbolsize=7) Plots(,x],i=1.Dimensions(t))], symbol=DIAMOND, symbolsize=8) ĭata Analysis: Fast Fourier Transform (FFT)Ī set of experimental numerical data can be analyzed with MATLAB® as a numerical engine. We must enforce the above variables as global in the MATLAB® environment so that the function mass_eqn can use them. The oscillator parameters of Mass M, Damping C, and Stiffness K are: Writeline(file, "function xdot=mass_eqn(t,x)"): Here, we create the file in the current directory. The MATLAB® function stored in the file mass_eqn.m can be created within Maple as follows, or it can be created by using any text editor. The simulation equations are coded as MATLAB® function files and are called from the Maple environment by using the ode45 command. Plots(P,heights=histogram,axes=boxed,labels=,title="temperature distribution") Ī simple spring-mass-dashpot is modeled as a second-order linear oscillator. Now, let us take the Maple solution vector and re-arrange its values in a 3 x 7 matrix that corresponds to the actual rectangular plate: Or, we can solve the system entirely within Maple: Now, we solve the system in the MATLAB® environment: , U edge at 100 units, and the other three edges at 0 temperature units. We have fixed the temperature along the U. , U ] ,we have the system AU=B, where the matrix A and column vector B encode the interconnectivity of the nodes and the profile of the boundary temperature:Ī := BandMatrix(,7,21,21,outputoptions=]):ī := Vector( 21,, datatype=float ) Representing the internal nodal temperatures U by the column vector. With our 21 plate-internal nodes and 20 boundary conditions ( = 7+3+7+3 ), the finite-difference 2-dimensional Laplace equation gives us an inhomogeneous linear system in 21 unknowns (the internal nodal temperatures). We model the plate as a 3 x 7 grid of nodes, where the nodes may be thought of as being interconnected with a square mesh of heat conductors. Heat Transfer: Finite Difference SolutionĪ difference equation method is used to find the static temperature distribution in a flat rectangular plate, given its boundary is held at a fixed temperature profile. The Eigenvectors are (in no particular order): The equivalent computations in the Maple environment: The Eigenvalues and Eigenvectors computed with MATLAB® are found: The Stiffness matrix K is tridiagonal with 2k on the center diagonal, and -k on the adjacent diagonals: To examine any of these Matrices, the Structured Data Browser can be used, by right-clicking the output Matrix and selecting Browse. M := DiagonalMatrix(, outputoptions=,storage=rectangular]) The mass matrix M is a matrix with m on the diagonal: The model equations may be formulated with the following matrix assignments. This formulation is used to compute the lowest natural frequencies and modes of a highly idealized 22-story building. Structural Analysis: A First Approximation Use the with command to access the functions in some useful packages by their short names:įor more information on the Maple-MATLAB® link, see Matlab.
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