EigenvaluesTutor - Maple Help
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Student[LinearAlgebra][EigenvaluesTutor] - interactive and step-by-step matrix eigenvalues

 Calling Sequence EigenvaluesTutor(M, opts)

Parameters

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

Description

 • The EigenvaluesTutor(M) command by default opens a Maplet window which allows you to work interactively through solving for the eigenvalues of M. Options provide other ways to show the step-by-step solutions, as described below.
 • The EigenvaluesTutor(M) command presents the techniques used in finding the eigenvalues of the square matrix $M$ by:
 1 Creating the matrix M - lambda*Id where Id is an identity matrix with dimensions equal to that of M
 2 Taking the determinant of M - lambda*Id
 3 Finding the roots of the resulting characteristic polynomial
 • 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 EigenvaluesTutor(M) command returns the eigenvalues as a column Vector.
 • 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)
 > $\mathrm{EigenvaluesTutor}\left(M\right)$
 > $\mathrm{EigenvaluesTutor}\left(M,\mathrm{output}=\mathrm{steps}\right)$
 $\begin{array}{lll}{}& {}& \text{Compute the Eigenvalues}\\ {}& {}& \left[\begin{array}{ccc}{1}& {2}& {0}\\ {2}& {3}& {2}\\ {0}& {2}& {1}\end{array}\right]\\ \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.}\\ {}& \text{◦}& \text{Use cofactor expansion on the}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}3\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{by}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}3\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{matrix}\\ {}& {}& \left({1}{-}{t}\right){\cdot }\left[\begin{array}{cc}{3}{-}{t}& {2}\\ {2}& {1}{-}{t}\end{array}\right]{+}\left({-1}\right){\cdot }{2}{\cdot }\left[\begin{array}{cc}{2}& {2}\\ {0}& {1}{-}{t}\end{array}\right]{+}{1}{\cdot }{0}{\cdot }\left[\begin{array}{cc}{2}& {3}{-}{t}\\ {0}& {2}\end{array}\right]\\ {}& \text{◦}& \text{Find the determinant of the 2 by 2 matrices by multiplying the diagonals}\\ {}& {}& \left({1}{-}{t}\right){\cdot }\left(\left({3}{-}{t}\right){\cdot }\left({1}{-}{t}\right){-}{2}{\cdot }{2}\right){+}\left({-1}\right){\cdot }{2}{\cdot }\left({2}{\cdot }\left({1}{-}{t}\right){-}{0}{\cdot }{2}\right){+}{1}{\cdot }{0}{\cdot }\left({2}{\cdot }{2}{-}{0}{\cdot }\left({3}{-}{t}\right)\right)\\ {}& \text{◦}& \text{Evaluate inside the brackets}\\ {}& {}& \left({1}{-}{t}\right){\cdot }\left(\left({3}{-}{t}\right){}\left({1}{-}{t}\right){-}{4}\right){+}\left({-1}\right){\cdot }{2}{\cdot }\left({2}{-}{2}{}{t}\right){+}{1}{\cdot }{0}{\cdot }{4}\\ {}& \text{◦}& \text{Multiply}\\ {}& {}& \left({1}{-}{t}\right){}\left(\left({3}{-}{t}\right){}\left({1}{-}{t}\right){-}{4}\right){+}\left({-}{4}{+}{4}{}{t}\right){+}{0}\\ {}& \text{◦}& \text{Evaluate}\\ {}& {}& \left({1}{-}{t}\right){}\left(\left({3}{-}{t}\right){}\left({1}{-}{t}\right){-}{4}\right){-}{4}{+}{4}{}{t}\\ {}& {}& \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]\end{array}$ (2)