Forcibly manipulating eigenvalues

Question

If eigenvalues of a matrix A are c1 and c2. Can I multiply or do any operations with the matrix A with something to manipulate the eigenvalues c1 and c2 to a desired value ?

Answer 1

Choose corresponding eigenvectors $v_1$ and $v_2$. Let $(\cdot,\cdot)$ be any scalar product with respect to which $v_1$ and $v_2$ are orthogonal to each other and both have norm one. Then define $$ A(s,t) = A + s(\cdot,v_1)v_1 + t(\cdot,v_2)v_2. $$ Then $A(s,t)v_1 = (c_1 + s)v_1$ and $A(s,t)v_2 = (c_2+t)v_2$. Hence $c_1+s$ and $c_2+t$ are eigenvalues of $A(s,t)$.

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To find such a scalar product, let $U = \operatorname{span}\{v_1,v_2\}$. Then every vector $v$ can be written as $x = \lambda_1v_1+\lambda_2v_2 + w$ with $\lambda_1,\lambda_2\in\mathbb R$ and $w\in U^\perp$. For $x = \lambda_1v_1+\lambda_2v_2 + w$ and $y = \mu_1v_1+\mu_2v_2 + z$ define $$ (x,y) := \lambda_1\mu_1 + \lambda_2\mu_2 + \langle w,z\rangle, $$ where $\langle\cdot,\cdot\rangle$ is the standard scalar product. Then $(\cdot,\cdot)$ is a scalar product as desired.