2

My matrix is

\begin{pmatrix} 3 & 4 & 5 \\ -2 & 7 & 3 \\ 5 & -8 & -3 \end{pmatrix}

Through the rule of Sarrus, I know (approximately) $\lambda_1 = 5.9$ $\lambda_2 = 3.5$ and $\lambda_3 = -2.4 $.

If someone could find $v_1$ I believe I can go from here to get the remaining $2$.

QuantumSuperfield
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MathsPro
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  • just compute $\ker(A-\lambda_i I_3)$ where $i=1,2,3$ ;-) – Surb May 06 '15 at 15:24
  • For each $\lambda$ write down the definition of an eigenvector. This is a set of simultaneous equations for the components of the eigenvector. – Paul May 06 '15 at 15:30
  • Sorry what is Ker? would you mind getting the vector for $lambda_1$ to give me an idea? – MathsPro May 06 '15 at 15:30
  • ker$(A - \lambda_{i} I) \vec v$ means solve $(A - \lambda_{i} I) \vec v = \vec 0$ – Matthew Cassell May 06 '15 at 15:38
  • I understand that if i do this i have 3 equations and 3 unknowns. How do i tackle this system of equations? – MathsPro May 06 '15 at 15:40
  • You might find this helpful: http://math.stackexchange.com/questions/989050/a-method-of-finding-the-eigenvector-that-i-dont-fully-understand – The Artist May 06 '15 at 15:41
  • i am trying to use method (2) but am having difficulty solving the system. I have read lots of info and all the textbooks seem to gloss over this stage as if it is obvious - but i need to see an example – MathsPro May 06 '15 at 15:43
  • @MathsPro Maybe it would help you to check out my answer to this question. –  May 06 '15 at 15:54
  • @MathsPro Do you know how to reduce a matrix (reduced row echelon form)? – Eff May 06 '15 at 15:56
  • @Eff Yes, i have three equations, 3 unkowns. if $v_1$ = $[ x:y:z]$ then putting them in row ecehlon form will give a solution for z? then y will be in terms of z, and x will be in terms of y and z? is this correct?

    I am getting a little confused because i though there was infinate solutions for eigen vectors?

     – MathsPro May 06 '15 at 17:16

2 Answers2

1

The eigenvectors will come from solving the equation $$\left[ \begin{array}{ccc} 3 & 4 & 5 \\ -2 & 7 & 3 \\ 5 & -8 & -3 \end{array}\right]\mathbf{v} = \lambda \mathbf{v}$$ for $\mathbf{v}$ where $$\mathbf{v}=\left[ \begin{array}{c} v_1 \\ v_2 \\ v_3 \end{array}\right]$$ Simply compute the matrix multiplication of $$\left[ \begin{array}{ccc} 3 & 4 & 5 \\ -2 & 7 & 3 \\ 5 & -8 & -3 \end{array}\right]\left[ \begin{array}{c} v_1 \\ v_2 \\ v_3 \end{array}\right]$$ and set it equal to $$\left[ \begin{array}{c} \lambda v_1 \\ \lambda v_2 \\ \lambda v_3 \end{array}\right]$$ to get a set of three equations. This will give you enough information to solve for $v_1,v_2$ and $v_3$, which will give you the eigenvector $\mathbf{v}$ that corresponds to the eigenvalue $\lambda$.

graydad
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  • Could you get $v_1$ to complete my understanding? – MathsPro May 06 '15 at 15:40
  • @MathsPro sorry, I do not want to calculate the matrix multiplication by hand. I can just tell you that you'll get something that looks like $$\left[ \begin{array}{ccc} av_1 +bv_2+ cv_3 \ dv_1 +ev_2+ fv_3 \ gv_1 +hv_2+ iv_3 \end{array}\right]$$ which gives you the equations $$\begin{array}{ccc} av_1 +bv_2+ cv_3 & = & \lambda v_1\ dv_1 +ev_2+ fv_3 & = & \lambda v_2\ gv_1 +hv_2+ iv_3 & = & \lambda v_3\end{array}$$ Three equations, three unknowns. You'll get $a$ through $i$ by doing the multiplication. Then plug in $\lambda$ and chug away. You could also use the rref tool on a TI calculator – graydad May 06 '15 at 15:46
1

Step 1. Find the characteristic polynomial $\phi(x)=\det(A-xI)=x^3-7x^2-2x+50.$

Step 2. Find the roots of $\phi(x).$ According to Wolfram Alpha, these are

enter image description here

enter image description here

and

enter image description here

These are the eigenvalues of the matrix. (note that despite the presence of $i$, these are all real numbers).

Step 3. For each eigenvalue $x$ from Step 2, solve the system of linear equations with matrix form $(A-xI)X=0$ (i.e. find the kernel (AKA nullspace) of $A-xI$). The solutions $X$ are the eigenvectors of $A$ for the eigenvalue $x.$

Step 4. Cry at the thought of even attempting Step 3.

Step 5. Check to see if you've got the matrix right. If no, start over with the correct matrix. If yes, get a computer to find eigenvalues and eigenvectors using software such as Matlab instead, or ask Wolfram Alpha to do that instead of asking it to find the characteristic polynomial. (I'm not aware of any simple methods for finding eigenvectors if you only know the approximate eigenvalues.)

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Randy E
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