My answer for eigenspace is valid right?

[Rijad Hadzic]

Homework Statement

Determine the characteristic polynomials, eigenvalues, and corresponding eigenspaces of the given 2x2 matricies

Homework Equations

The Attempt at a Solution

<br /> <br /> \begin{pmatrix}<br /> 5 &amp; 2\\<br /> -8 &amp; -3 \\<br /> \end{pmatrix}<br />
thus

<br /> <br /> \begin{pmatrix}<br /> 5-\lambda &amp; 2\\<br /> -8 &amp; -3-\lambda \\<br /> \end{pmatrix}<br />

determinant is = to: \lambda^2 -2\lambda + 1

which gives value lambda = 1

plugging into <br /> <br /> \begin{pmatrix}<br /> 5-\lambda &amp; 2\\<br /> -8 &amp; -3-\lambda \\<br /> \end{pmatrix}<br />

you get<br /> <br /> \begin{pmatrix}<br /> 4 &amp; 2\\<br /> -8 &amp; -4 \\<br /> \end{pmatrix}<br />

using rref you get<br /> <br /> \begin{pmatrix}<br /> 1 &amp; .5\\<br /> 0 &amp; 0\\<br /> \end{pmatrix}<br />

setting x2 = r, I get eigenspace r*<br /> <br /> \begin{pmatrix}<br /> -1/2\\<br /> 1 \\<br /> \end{pmatrix}<br />

but my book is telling me the anser is r*<br /> <br /> \begin{pmatrix}<br /> 1\\<br /> -2 \\<br /> \end{pmatrix}<br />

our answers are the same thing right?

 

[Dick]

Homework Statement

Determine the characteristic polynomials, eigenvalues, and corresponding eigenspaces of the given 2x2 matricies

Homework Equations

The Attempt at a Solution

<br /> <br /> \begin{pmatrix}<br /> 5 &amp; 2\\<br /> -8 &amp; -3 \\<br /> \end{pmatrix}<br />
thus

<br /> <br /> \begin{pmatrix}<br /> 5-\lambda &amp; 2\\<br /> -8 &amp; -3-\lambda \\<br /> \end{pmatrix}<br />

determinant is = to: \lambda^2 -2\lambda + 1

which gives value lambda = 1

plugging into<br /> <br /> \begin{pmatrix}<br /> 5-\lambda &amp; 2\\<br /> -8 &amp; -3-\lambda \\<br /> \end{pmatrix}<br />

you get<br /> <br /> \begin{pmatrix}<br /> 4 &amp; 2\\<br /> -8 &amp; -4 \\<br /> \end{pmatrix}<br />

using rref you get<br /> <br /> \begin{pmatrix}<br /> 1 &amp; .5\\<br /> 0 &amp; 0\\<br /> \end{pmatrix}<br />

setting x2 = r, I get eigenspacer*<br /> <br /> \begin{pmatrix}<br /> -1/2\\<br /> 1 \\<br /> \end{pmatrix}<br />

but my book is telling me the anser isr*<br /> <br /> \begin{pmatrix}<br /> 1\\<br /> -2 \\<br /> \end{pmatrix}<br />

our answers are the same thing right?

Of course they are. If $v$ is an eigenvector then so is $cv$ for any nonzero $c$. Your values of $r$ just differ by a factor of -2.

 

[Rijad Hadzic]

Of course they are. If $v$ is an eigenvector then so is $cv$ for any nonzero $c$. Your values of $r$ just differ by a factor of -2.

alright ty was just making sure. I do overthink small things like this but your explanation makes sense.

 

[Ray Vickson]

Homework Statement

Determine the characteristic polynomials, eigenvalues, and corresponding eigenspaces of the given 2x2 matricies

Homework Equations

The Attempt at a Solution

<br /> <br /> \begin{pmatrix}<br /> 5 &amp; 2\\<br /> -8 &amp; -3 \\<br /> \end{pmatrix}<br />
thus

<br /> <br /> \begin{pmatrix}<br /> 5-\lambda &amp; 2\\<br /> -8 &amp; -3-\lambda \\<br /> \end{pmatrix}<br />

determinant is = to: \lambda^2 -2\lambda + 1

which gives value lambda = 1

plugging into<br /> <br /> \begin{pmatrix}<br /> 5-\lambda &amp; 2\\<br /> -8 &amp; -3-\lambda \\<br /> \end{pmatrix}<br />

you get<br /> <br /> \begin{pmatrix}<br /> 4 &amp; 2\\<br /> -8 &amp; -4 \\<br /> \end{pmatrix}<br />

using rref you get<br /> <br /> \begin{pmatrix}<br /> 1 &amp; .5\\<br /> 0 &amp; 0\\<br /> \end{pmatrix}<br />

setting x2 = r, I get eigenspacer*<br /> <br /> \begin{pmatrix}<br /> -1/2\\<br /> 1 \\<br /> \end{pmatrix}<br />

but my book is telling me the anser isr*<br /> <br /> \begin{pmatrix}<br /> 1\\<br /> -2 \\<br /> \end{pmatrix}<br />

our answers are the same thing right?

You should point out that "1" is a double eigenvalue; that is, the eigenvalues of the matrix are 1,1. You might also want to point out that this matrix is "deficient": its "eigenspace" has only one dimension and so does not span the whole space. (It I were marking this question I would give full points only if the student mentioned those things---but of course, I am not marking it.)

 

[Mark44]

Thread moved to Calculus section. Questions about eigen-<whatever> are well beyond precalculus, IMO.