From Strang's Introduction to Linear Algebra (p375), there is a paragraph on the instability of eigenvalues in relation to the stability of singular values when $A$ is altered slightly. Moreover, it says, that the instability of eigenvalues is only if $AA^T$ is far from $A^TA$, but if those are very similar, the eigenvalues are also stable. The book goes on to present a $4\times4$ example with one value altered by $1/60000$ which changes the eigenvalues by $1/10$ but singular value change is only $1/60000$.

To put what I understood from this claim and the example in definite terms, the eigenvalues of $A$ are seriously unstable if $A^TA$ is significantly different $AA^T$, i.e. on altering $a_{ij}$ in $A$ by $\Delta a_{ij}$, the $\lambda_A$ change by values much greater ($\Delta \lambda_A >> \Delta a_{ij}$) but singular values change in the same order ($\Delta \sigma_A \approx\Delta a_{ij}$).

The overarching question I have is, I do not understand, intuitively or mathematically, how this came to be

My attempts:

I tried to characterize $\Delta \lambda_A$ by looking at how the coefficient of the polynomial $det(A-\lambda_A I) = 0$ change when $a_{ij}$ becomes $a_{ij} + \Delta a_{ij}$. But if I were to compare this to the singular value case, I would be looking at the change in coefficients of the polynomial $det(A^TA - \sigma_A^2I) = 0$, but in $A^TA$, the $\Delta a_{ij}$ term would be actually present at more than one entries and it would seem that it would actually lead to a greater change in the coefficients! I also am not sure how would a change in coefficients be related to the change in the zeros of the polynomial.

Furthermore, why would the relation between $AA^T$ and $A^TA$ affect $\Delta \lambda_A$ rather than $\Delta \sigma_A$, as one would expect since the singular value is quite strongly associated with the eigenvectors of $AA^T$ and $A^TA$, but the eigenvalues of $A$ appears to not have a direct relation with them.

Any more clarity on the intuition of the three bold statements would be appreciated.


Let $A\in M_n(\mathbb{R})$ and $\lambda$ be a SIMPLE real eigenvalue of $A$. Let $Ax=\lambda x$ where $||x||^2=1$. Then $x^Tx=1,x^T(\Delta x)=0$.

$(\Delta A)x+A(\Delta x)=(\Delta \lambda)x+\lambda (\Delta x)$ implies that

(*) $x^T(\Delta A)x+x^TA(\Delta x)=\Delta \lambda$ and

(*bis) $x^T(\Delta A)x=\Delta \lambda$ when $AA^T=A^TA$.

Let $AA^T=S$ (a symmetric matrix) and $Sy=\sigma y$, where $||y||^2=1$. Then

$(\Delta S)y+S(\Delta y)=(\Delta \sigma)y+\sigma (\Delta y)$ implies that

(**) $y^T(\Delta S)y=\Delta \sigma$.

Then $\Delta \sigma$ has same order as $\Delta S=(\Delta A)A^T+A(\Delta A)^T$, that is, same order as $\Delta A$.

On the other hand, $\Delta \lambda$ has same order as $\max(order(\Delta A),order(\Delta x))$ and as $order(\Delta A)$ when $AA^T=A^TA$.

Finally a matrix $A$ with instable eigenvalue $\lambda$ is obtained when $\Delta x$ is very large. The Strang's example is

$A=\begin{pmatrix}0&1&0&0\\0&0&2&0\\0&0&0&3\\0&0&0&0\end{pmatrix}$ and $\Delta A=\begin{pmatrix}0&0&0&0\\0&0&0&0\\0&0&0&0\\1/60000&0&0&0\end{pmatrix}$.

Note that $0$ is not a simple eigenvalue; yet instability is further caused by the magnitude of $\Delta x$ (the unique eigenvector of $A$ explodes in $4$ eigenvectors that are far from the first one -when we change $a_{4,1}=0$ into $1/60000$-).

EDIT. To get the simplicity of the eigenvalues (as I suppose it), just choose $a_{4,1} = 10^{-10}$ (then there are $4$ distinct eigenvalues) and change it to $a_{4,1}=1/60000$.

Then check that the eigenvector associated with the $> 0$ eigenvalue "moves" quickly. It's your business.

  • $\begingroup$ Sorry for being slow, but can you expand/explain a bit on some of the statements and how they came about, and how one implies the other (I understood some but not most)? Especially the first statement with $\Delta$s? $\endgroup$ Mar 29 '20 at 9:37
  • $\begingroup$ It's the standard derivation; for example $\dfrac {\partial A}{\partial a_{i,j}}\approx \dfrac{\Delta A}{\Delta a_{i,j}}$; moreover , I write only the numerators -it is a notation widely used in physics- $\endgroup$
    – user91684
    Mar 29 '20 at 11:13
  • $\begingroup$ Can you expand on how $xT(\Delta A)x+xTA(\Delta x)=\Delta λ$ and $yT(\Delta S)y+yTS(\Delta y)=\Delta σ$ lead to different conclusions despite having similar forms? (in the former, in general, we do not remove the contribution of $\Delta x$ while in the latter we remove the contribution of $\Delta y$)? And by a simple eigenvalue, what do you mean? $\endgroup$ May 9 '20 at 9:06

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