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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 tiId+M=0 for each eigenvalue ti

• 

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

withStudentLinearAlgebra:

M1,2,0|2,3,2|0,2,1

M120232021

(1)

EigenvectorsTutorM

EigenvectorsTutorM,output=steps

Compute the eigenvectors120232021Compute the eigenvaluesCalculate A=M-t*Id1t2023t2021tFind the determinant; this is also called the characteristic polynomial of M.t3+5t2+t5Solve; the eigenvalues are the roots of the characteristic polynomial.51−1Select an Eigenvalue1Subtract the eigenvalue times the identity matrix from MCalculate A=M-tId020222020Solve the system of equations AX=0=0Apply Gaussian Elimination Swap rows 1 and 2222020020 Subtract 1 times row 2 from row 3; (R3 = R3-1*R2)222020000This is the solution to the system of equationsX=This is an eigenvector−101Select an Eigenvalue−1Subtract the eigenvalue times the identity matrix from MCalculate A=M-tId220242022Solve the system of equations AX=0=0Apply Gaussian Elimination Subtract 1 times row 1 from row 2; (R2 = R2-1*R1)220022022 Subtract 1 times row 2 from row 3; (R3 = R3-1*R2)220022000This is the solution to the system of equationsX=This is an eigenvector1−11Select an Eigenvalue5Subtract the eigenvalue times the identity matrix from MCalculate A=M-tId−4202−2202−4Solve the system of equations AX=0=0Apply Gaussian Elimination Add 1/2 times row 1 to row 2; (R2 = 1/2*R1+R2)−4200−1202−4 Add 2 times row 2 to row 3; (R3 = 2*R2+R3)−4200−12000This is the solution to the system of equationsX=This is an eigenvector121

(2)

See Also

Student[LinearAlgebra], Student[LinearAlgebra][Determinant], Student[LinearAlgebra][Eigenvalues], Student[LinearAlgebra][EigenvaluesTutor], Student[LinearAlgebra][Eigenvectors], Student[LinearAlgebra][LinearSolve], Student[LinearAlgebra][LinearSolveTutor]