Computing Eigen-coordinates

A very short guide

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Topics Covered



How to Compute Eigen-coordinates

In every example so far, the components of each vector along the eigen-axes have been computed for you. This was done to allow you to focus on how they work, rather than you habitually stopping your thoughts so you can calulate them. But of course, one must learn how to calculate them. Given the coordinates (x,y)(x, y) for our column vector using the regular axes, we need to find new coordinates x′,y′x', y' with the property that

x′v⃗1+y′v⃗2=[xy].x' \vec{v}_1 + y' \vec{v}_2 = \begin{bmatrix} x \\ y \end{bmatrix}.

This can be solved very quickly with a special matrix PP with columns v⃗1\vec{v}_1 and v⃗2\vec{v}_2:

[↑v⃗1↓↑v⃗2↓][x′y′]=[xy] \begin{bmatrix} \begin{array}{c} \uparrow \\ \vec{v}_1 \\ \downarrow \end{array} & \begin{array}{c} \uparrow \\ \vec{v}_2 \\ \downarrow \end{array} \end{bmatrix} \begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} x \\ y \end{bmatrix}

[x′y′]=[↑v⃗1↓↑v⃗2↓]−1[xy] \begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} \begin{array}{c} \uparrow \\ \vec{v}_1 \\ \downarrow \end{array} & \begin{array}{c} \uparrow \\ \vec{v}_2 \\ \downarrow \end{array} \end{bmatrix}^{-1} \begin{bmatrix} x \\ y \end{bmatrix}

And there you go! Take your matrix of eigenvectors, find its inverse, and apply it to your standard coordinates.



Problem 1: Computing Eigen-coordinates

Consider the eigenvector basis

v⃗1=[11],v⃗2=[−11].\vec{v}_1 = \begin{bmatrix} 1 \\ 1 \end{bmatrix},\qquad \vec{v}_2 = \begin{bmatrix} -1 \\ 1 \end{bmatrix}.

Find the eigen-coordinates (x′,y′)(x',y') of

v⃗=[42].\vec{v} = \begin{bmatrix} 4 \\ 2 \end{bmatrix}.

In other words, find x′x' and y′y' such that

x′v⃗1+y′v⃗2=v⃗.x'\vec{v}_1+y'\vec{v}_2=\vec{v}.

Enter two numbers in order: first x′, then y′.

Answer the question correctly to reveal the short diagonalization section and finish this page.