EigenvectorsTutor - Maple Help

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Student[LinearAlgebra][EigenvectorsTutor] - interactive and step-by-step matrix eigenvectors

 Calling Sequence EigenvectorsTutor(M, opts)

Parameters

 M - square Matrix opts - (optional) equation(s) of the form option=value where equation is output or displaystyle

Description

 • The EigenvectorsTutor(M) command by default opens a Maplet window which allows you to work interactively through solving for the eigenvectors of M. Options provide other ways to show the step-by-step solutions, as described below.
 • The EigenvectorsTutor(M) command presents the techniques used in finding the eigenvectors of the square matrix $M$ by:
 1 Finding the eigenvalues
 2 Solving the equation $-{t}_{i}\mathrm{Id}+M=0$ for each eigenvalue ${t}_{i}$
 • The Matrix M must be square and of dimension 4 at most.
 • Floating-point numbers in M are converted to rationals before computation begins.
 • If the symbolic expression representing an eigenvalue grows too large, then the value displayed in the Maplet application window is a floating-point approximation to it (obtained by applying evalf).  The underlying computations continue to be performed using exact arithmetic, however.
 • The EigenvectorsTutor(M) command returns the eigenvectors as a set of column Vectors.
 • The following options can be used to control how the problem is displayed and what output is returned, giving the ability to generate step-by-step solutions directly without going through the Maplet tutor interface:
 – output = steps,canvas,script,record,list,print,printf,typeset,link (default: maplet)

The output options are described in Student:-Basics:-OutputStepsRecord.  Use output = steps to get the default settings for displaying step-by-step solution output.

 – displaystyle= columns,compact,linear,brief (default: linear)

The displaystyle options are described in Student:-Basics:-OutputStepsRecord.

Examples

 > $\mathrm{with}\left(\mathrm{Student}\left[\mathrm{LinearAlgebra}\right]\right):$
 > $M≔⟨⟨1,2,0⟩|⟨2,3,2⟩|⟨0,2,1⟩⟩$
 ${M}{≔}\left[\begin{array}{ccc}{1}& {2}& {0}\\ {2}& {3}& {2}\\ {0}& {2}& {1}\end{array}\right]$ (1)
 > $\mathrm{EigenvectorsTutor}\left(M\right)$
 > $\mathrm{EigenvectorsTutor}\left(M,\mathrm{output}=\mathrm{steps}\right)$
 $\begin{array}{lll}{}& {}& \text{Compute the eigenvectors}\\ {}& {}& \left[\begin{array}{ccc}{1}& {2}& {0}\\ {2}& {3}& {2}\\ {0}& {2}& {1}\end{array}\right]\\ \text{▫}& {}& \text{Compute the eigenvalues}\\ {}& \text{◦}& \text{Calculate A=M-t*Id}\\ {}& {}& \left[\begin{array}{ccc}{1}{-}{t}& {2}& {0}\\ {2}& {3}{-}{t}& {2}\\ {0}& {2}& {1}{-}{t}\end{array}\right]\\ {}& \text{◦}& \text{Find the determinant; this is also called the characteristic polynomial of M.}\\ {}& {}& {-}{{t}}^{{3}}{+}{5}{}{{t}}^{{2}}{+}{t}{-}{5}\\ {}& \text{◦}& \text{Solve; the eigenvalues are the roots of the characteristic polynomial.}\\ {}& {}& \left[\begin{array}{c}{5}\\ {1}\\ {-1}\end{array}\right]\\ \text{•}& {}& \text{Select an Eigenvalue}\\ {}& {}& {1}\\ \text{•}& {}& \text{Subtract the eigenvalue times the identity matrix from M}\\ {}& {}& \left[{}\right]\\ \text{•}& {}& \text{Calculate A=M-tId}\\ {}& {}& \left[\begin{array}{ccc}{0}& {2}& {0}\\ {2}& {2}& {2}\\ {0}& {2}& {0}\end{array}\right]\\ \text{•}& {}& \text{Solve the system of equations AX=0}\\ {}& {}& \left[{}\right]{=}{0}\\ \text{▫}& {}& \text{Apply Gaussian Elimination}\\ {}& \text{◦}& \text{Swap rows 1 and 2}\\ {}& {}& \left[\begin{array}{ccc}{2}& {2}& {2}\\ {0}& {2}& {0}\\ {0}& {2}& {0}\end{array}\right]\\ {}& \text{◦}& \text{Subtract 1 times row 2 from row 3; (R3 = R3-1*R2)}\\ {}& {}& \left[\begin{array}{ccc}{2}& {2}& {2}\\ {0}& {2}& {0}\\ {0}& {0}& {0}\end{array}\right]\\ \text{•}& {}& \text{This is the solution to the system of equations}\\ {}& {}& {X}{=}\left[{}\right]\\ \text{•}& {}& \text{This is an eigenvector}\\ {}& {}& \left[\begin{array}{c}{-1}\\ {0}\\ {1}\end{array}\right]\\ \text{•}& {}& \text{Select an Eigenvalue}\\ {}& {}& {-1}\\ \text{•}& {}& \text{Subtract the eigenvalue times the identity matrix from M}\\ {}& {}& \left[{}\right]\\ \text{•}& {}& \text{Calculate A=M-tId}\\ {}& {}& \left[\begin{array}{ccc}{2}& {2}& {0}\\ {2}& {4}& {2}\\ {0}& {2}& {2}\end{array}\right]\\ \text{•}& {}& \text{Solve the system of equations AX=0}\\ {}& {}& \left[{}\right]{=}{0}\\ \text{▫}& {}& \text{Apply Gaussian Elimination}\\ {}& \text{◦}& \text{Subtract 1 times row 1 from row 2; (R2 = R2-1*R1)}\\ {}& {}& \left[\begin{array}{ccc}{2}& {2}& {0}\\ {0}& {2}& {2}\\ {0}& {2}& {2}\end{array}\right]\\ {}& \text{◦}& \text{Subtract 1 times row 2 from row 3; (R3 = R3-1*R2)}\\ {}& {}& \left[\begin{array}{ccc}{2}& {2}& {0}\\ {0}& {2}& {2}\\ {0}& {0}& {0}\end{array}\right]\\ \text{•}& {}& \text{This is the solution to the system of equations}\\ {}& {}& {X}{=}\left[{}\right]\\ \text{•}& {}& \text{This is an eigenvector}\\ {}& {}& \left[\begin{array}{c}{1}\\ {-1}\\ {1}\end{array}\right]\\ \text{•}& {}& \text{Select an Eigenvalue}\\ {}& {}& {5}\\ \text{•}& {}& \text{Subtract the eigenvalue times the identity matrix from M}\\ {}& {}& \left[{}\right]\\ \text{•}& {}& \text{Calculate A=M-tId}\\ {}& {}& \left[\begin{array}{ccc}{-4}& {2}& {0}\\ {2}& {-2}& {2}\\ {0}& {2}& {-4}\end{array}\right]\\ \text{•}& {}& \text{Solve the system of equations AX=0}\\ {}& {}& \left[{}\right]{=}{0}\\ \text{▫}& {}& \text{Apply Gaussian Elimination}\\ {}& \text{◦}& \text{Add 1/2 times row 1 to row 2; (R2 = 1/2*R1+R2)}\\ {}& {}& \left[\begin{array}{ccc}{-4}& {2}& {0}\\ {0}& {-1}& {2}\\ {0}& {2}& {-4}\end{array}\right]\\ {}& \text{◦}& \text{Add 2 times row 2 to row 3; (R3 = 2*R2+R3)}\\ {}& {}& \left[\begin{array}{ccc}{-4}& {2}& {0}\\ {0}& {-1}& {2}\\ {0}& {0}& {0}\end{array}\right]\\ \text{•}& {}& \text{This is the solution to the system of equations}\\ {}& {}& {X}{=}\left[{}\right]\\ \text{•}& {}& \text{This is an eigenvector}\\ {}& {}& \left[\begin{array}{c}{1}\\ {2}\\ {1}\end{array}\right]\end{array}$ (2)