Linear Algebra Done Right · Problem 5D.23

The key insight is that eigenvalues of a matrix depend only on the matrix itself, not on whether you organize information by rows or columns. The standard Gershgorin theorem uses row sums because it applies to $A\mathbf{x} = \lambda\mathbf{x}$, but you can apply the same reasoning to the transpose: if $\lambda$ is an eigenvalue of $A$, it's also an eigenvalue of $A^T$. By applying the original row-based Gershgorin theorem to $A^T$ (where the rows of $A^T$ are the columns of $A$), you get disks defined by column sums of $A$—proving the theorem holds with this switched definition.

The key insight is that eigenvalues of a matrix depend only on the matrix itself, not on whether you organize information by rows or columns. The standard Gershgorin theorem uses row sums because it applies to $A\mathbf{x} = \lambda\mathbf{x}$, but you can apply the same reasoning to the transpose: if $\lambda$ is an eigenvalue of $A$, it's also an eigenvalue of $A^T$. By applying the original row-based Gershgorin theorem to $A^T$ (where the rows of $A^T$ are the columns of $A$), you get disks defined by column sums of $A$—proving the theorem holds with this switched definition.