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:

Finding the eigenvalues

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)

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

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{LinearAlgebra}\right]\right)\:$

$M≔⟨⟨1\,2\,0⟩\left|⟨2\,3\,2⟩\right|⟨0\,2\,1⟩⟩$

$M≔\left[\begin{array}{ccc}1& 2& 0\\ 2& 3& 2\\ 0& 2& 1\end{array}\right]$

$\mathrm{EigenvectorsTutor}\left(M\right)$

$\mathrm{EigenvectorsTutor}\left(M\,\mathrm{output}=\mathrm{steps}\right)$

$\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+5t^2+t-5\\ & \text{◦}& \text{Solve; the eigenvalues are the roots of the characteristic polynomial.}\\ & & \left[\begin{array}{c}5\\ 1\\ \mathrm{-1}\end{array}\right]\\ \text{•}& & \text{Select an Eigenvalue}\\ & & 1\\ \text{•}& & \text{Subtract the eigenvalue times the identity matrix from M}\\ & & \left[\begin{array}{ccc}1& 2& 0\\ 2& 3& 2\\ 0& 2& 1\end{array}\right]-1\cdot \left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\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[\begin{array}{ccc}0& 2& 0\\ 2& 2& 2\\ 0& 2& 0\end{array}\right]\cdot X=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=c\cdot \left[\begin{array}{c}\mathrm{-1}\\ 0\\ 1\end{array}\right]\\ \text{•}& & \text{This is an eigenvector}\\ & & \left[\begin{array}{c}\mathrm{-1}\\ 0\\ 1\end{array}\right]\\ \text{•}& & \text{Select an Eigenvalue}\\ & & \mathrm{-1}\\ \text{•}& & \text{Subtract the eigenvalue times the identity matrix from M}\\ & & \left[\begin{array}{ccc}1& 2& 0\\ 2& 3& 2\\ 0& 2& 1\end{array}\right]-\left(\mathrm{-1}\right)\cdot \left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\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[\begin{array}{ccc}2& 2& 0\\ 2& 4& 2\\ 0& 2& 2\end{array}\right]\cdot X=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=c\cdot \left[\begin{array}{c}1\\ \mathrm{-1}\\ 1\end{array}\right]\\ \text{•}& & \text{This is an eigenvector}\\ & & \left[\begin{array}{c}1\\ \mathrm{-1}\\ 1\end{array}\right]\\ \text{•}& & \text{Select an Eigenvalue}\\ & & 5\\ \text{•}& & \text{Subtract the eigenvalue times the identity matrix from M}\\ & & \left[\begin{array}{ccc}1& 2& 0\\ 2& 3& 2\\ 0& 2& 1\end{array}\right]-5\cdot \left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\\ \text{•}& & \text{Calculate A=M-tId}\\ & & \left[\begin{array}{ccc}\mathrm{-4}& 2& 0\\ 2& \mathrm{-2}& 2\\ 0& 2& \mathrm{-4}\end{array}\right]\\ \text{•}& & \text{Solve the system of equations AX=0}\\ & & \left[\begin{array}{ccc}\mathrm{-4}& 2& 0\\ 2& \mathrm{-2}& 2\\ 0& 2& \mathrm{-4}\end{array}\right]\cdot X=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}\mathrm{-4}& 2& 0\\ 0& \mathrm{-1}& 2\\ 0& 2& \mathrm{-4}\end{array}\right]\\ & \text{◦}& \text{Add 2 times row 2 to row 3; (R3 = 2*R2+R3)}\\ & & \left[\begin{array}{ccc}\mathrm{-4}& 2& 0\\ 0& \mathrm{-1}& 2\\ 0& 0& 0\end{array}\right]\\ \text{•}& & \text{This is the solution to the system of equations}\\ & & X=c\cdot \left[\begin{array}{c}1\\ 2\\ 1\end{array}\right]\\ \text{•}& & \text{This is an eigenvector}\\ & & \left[\begin{array}{c}1\\ 2\\ 1\end{array}\right]\end{array}$