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I understand that if A is a matrix then there may exist a vector v that A scales.

What I don't understand is why this is useful and why is it used for e.g orthogonal bases. What's the point of knowing what A scales if it only works for a certain couple of vectors? Doesn't it have to apply to every vector to be able to tell something about A or be able to use it in general?

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  • If $Av_1=\lambda_1v_1$ and $Av_2=\lambda_2v_2$ and $v=bv_1+cv_2$, then $A(v)=A(bv_1+cv_2)=bA(v_1)+cA(v_2)=b\lambda_1v_1+c\lambda_2v_2$ – J. W. Tanner Mar 13 '20 at 03:13
  • They are useful for many purposes, but one of the most basic things is diagonalization. Diagonal matrices are much easier to deal with, and if you have a basis of eigenvectors of $A$ then $A$ is diagonal in that basis. For example, it's much easier to compute $A^7$ for diagonal $A$: just raise each diagonal entry to the power of $7$. – Jair Taylor Mar 13 '20 at 03:16
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    See for example https://math.stackexchange.com/questions/26662 for several examples in physics – Andrei Mar 13 '20 at 03:23
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    See this too – J. W. Tanner Mar 13 '20 at 03:26

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