Let's say I have a matrix M such that for vectors R and r in xy-coordinate system:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]R=Mr[/tex]

Suppose we diagonalized it so that there is another matrix D such that for vectors R' (which is also R) and r' (which is also r) in x'y'-coordinate system:

[tex]R'=Dr'[/tex]

D is a matrix with zero elements except for its main diagonal, also, these elements are the eigenvalues of matrix M. The order of these values in the diagonal are supposed to be arbitrary, as what my textbook says.

Let's look at the case of 2D vectors and in which the eigenvalues are perpendicular to each other (thus there is rotation of the original xy axes to x'y' axes by some value [itex]\theta[/itex]), let's say the matrix D operates on r' such that the components of r' transforms to R' in a way that:

[tex]X'=x'\ and\ Y'=6y'[/tex]

Hence M acts on vectors such that it 'streches' them to the direction of y'.

My question is, I am told that the choice of the order of the eigenvalues in a diagonalized matrix is arbitrary, and thus the choice of which of the eigenvectors corresponds to x' and y' axes are also arbitrary. Are we supposed to just examine the behavior of the vectors in the xy-coordinate system so that we'd know which of the eigenvectors would be parallel to x' or y'? For this example would the vectors' component along the angle [itex]\theta [/itex] degrees be the one multiplied by 6 or is it along the one along the direction [itex]\theta + \frac{\Pi}{2}[/itex] from the x-axis?

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# Diagonalization question

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