I have the following problem: let $A$ be an $m \times m$ matrix that is not symmetric and real. Then, I am interested in its eigenspaces that correspond to the largest eigenvalue, $\lambda_{\text{max}}$. Without affecting the column span of the eigenspace associated with the largest root, say $\text{sp}\,r_{\text{max}}$, what can I do to $A$ to effect the following change: 1. remove $\lambda_{\text{max}}$, 2. insert $\mu$ in its stead, 3. obtain the relation $A\, r_{\text{max}} = \mu r_{\text{max}}$ so that the eigenvalue associated with $\text{sp}\,r_{\text{max}}$ is now $\mu$. I was thinking of possibly using the real Schur decomposition, thanks to its numerical stability. Any help is much appreciated!
Follow-up: suppose that the eigenvalue is potentially complex but the matrix is real. Then, as suggested, calculating $A + I \left(\mu -\lambda_{\text{max}} \right)$ would cause a problem because $\lambda_{\text{max}}$ comes in a conjugate pair. We (re-)define the notion of 'max' to be maximal by modulus, so $\lambda_{\text{max}}$ is such that $||\lambda_{\text{max}}||$ is maximal. (This arises from the fact that I am estimating $A$ using least squares. A few references, avenues I have considered is applying the method of Gershgorin discs or the Bauer-Fike theorem just to bound $||\lambda_{\text{max}}||$ using the (real) entries of $A$ but have not made much progress.
Moreover, it would be interesting to know if one could use the derivative of $A$ with respect to its eigenvalues to 'slide it' towards a matrix that would have the desired root, very much one uses the first derivative of a function to attain a linear approximation to.
The "obvious" thing to do is to change each $\lambda_{max}$ on the diagonal of the Schur decomposition to $\mu$, but that may change the eigenspace. For example, if $$A = \pmatrix{1 & 1\cr 0 & 2\cr}$$ (already upper triangular) the eigenspace for $2$ is spanned by $\pmatrix{1\cr 1\cr}$, but if you change the $2$ to $\mu$ the eigenspace for $\mu$ becomes $\pmatrix{1/(\mu-1)\cr 1\cr}$.
Instead, why not just add $(\mu - \lambda_{max}) I$ to the matrix? The eigenspaces stay the same, just that all eigenvalues are shifted by $\mu - \lambda_{max}$.
EDIT: Another possibility is to use the functional calculus: for any polynomial $p(X)$, $p(A)$ is a matrix whose eigenvalues are $p(\lambda)$ for eigenvalues $\lambda$ of $A$. If $v$ is an eigenvector of $A$ for $\lambda$, then $p(A) v = p(\lambda) v$. Moreover, if $p$ has real coefficients, $p(A)$ will still be real. Choose a polynomial such that $p(\lambda_{max}) = \mu$.