Eigenvalues, Geo/Alg Multiplicity of a matrix filled with the same number I was working on a practice exam for my Linear Algebra final and was confused by this concept. I know that for a square matrix of all 1's, the eigenvalues should be 0 and n and the algebraic multiplicity should be 1 and n-1 (with n being the number of rows/cols).
How does this change for a matrix filled with the number 2019 (rather than a matrix of 1's)?
Also, I do not understand the difference between algebraic and geometric multiplicity and how that would be different for a matrix of all 1's vs a matrix of all 2019's.
Below is the question and the solutions.
Thanks!


 A: Let $J$ denote the matrix with all $1$s.
If you have an equation $Jv = \lambda v$ then you have this equation as well: $(2019J)v = (2019\lambda)v$. So if $v$ is an eigenvector of $J$ with eigenvalue $\lambda$, then $v$ is still an eigenvector of $2019J$ and now with eigenvalue $2019\lambda$. (And as you said, the eigenvalues of $J$ are $\lambda = 0$ and $\lambda = n = 2019$.)
The algebraic multiplicity of an eigenvalue is the number of times it occurs as a root of the characteristic polynomial. Here, as you said, $\det(\lambda I - J) = \lambda^{n-1}(\lambda - 1)$ (algebraic multiplicites $n-1$ and $1$). Now consider $\det(2019\lambda I - 2019 J) = 2019^n \det(\lambda I - J)$ and you can make a substitution $\lambda' = 2019\lambda$ (and $\lambda = \lambda'/2019$ to get $\det(\lambda' I - 2019 J)$. If you put that all together you will obtain the characteristic polynomial of $2019J$.
The geometric multiplicity of an eigenvalue is the dimension of $\operatorname{Null}(A - \lambda I)$. That is, the geometric multiplicity is the maximum number $m$ such that there exists $m$ linearly independent eigenvectors $v_1,\dots,v_m$ (which will form a basis for $\operatorname{Null}(A - \lambda I)$). By the first paragraph, $2019Jv_i = (2019\lambda)v_i$ for $i = 1,\dots,m$. Or another way to describe this is $\operatorname{Null}(J - \lambda I) = \operatorname{Null}(2019[J - \lambda I])$.
