# Levy-Desplanques theorem

A strictly diagonally dominant matrix is non-singular. In other words, let $A\in\mathbf{C}^{n,n}$ be a matrix satisfying the property

 $\left|a_{ii}\right|>\sum_{j\neq i}\left|a_{ij}\right|\qquad\forall i;$

then $\det(A)\neq 0$.

Proof: Let $\det(A)=0$; then a non-zero vector $\mathbf{x}$ exists such that $A\mathbf{x}=\mathbf{0}$; let $M$ be the index such that $\left|x_{M}\right|=\max(\left|x_{1}\right|,\left|x_{2}\right|,\cdots,\left|x_{% n}\right|)$, so that $\left|x_{j}\right|\leq\left|x_{M}\right|\quad\forall j$; we have

$a_{M1}x_{1}+a_{M2}x_{2}+\cdots+a_{MM}x_{M}+\cdots+a_{Mn}x_{n}=0$

which implies:

$\left|a_{MM}\right|\left|x_{M}\right|=\left|a_{MM}x_{M}\right|=\left|\sum_{j% \neq M}a_{Mj}x_{j}\right|\leq\sum_{j\neq M}\left|a_{Mj}\right|\left|x_{j}% \right|\leq\left|x_{M}\right|\sum_{j\neq M}\left|a_{Mj}\right|$

that is

$\left|a_{MM}\right|\leq\sum_{j\neq M}\left|a_{Mj}\right|,$

in contrast with strictly diagonally dominance definition.$\square$

Remark: the Levy-Desplanques theorem is equivalent to the well-known Gerschgorin circle theorem. In fact, let’s assume Levy-Desplanques theorem is true, and let $A$ a $n\times n$ complex-valued matrix, with an eigenvalue $\lambda$; let’s apply Levy-Desplanques theorem to the matrix $B=A-\lambda I$, which is singular by definition of eigenvalue: an index $i$ must exist for which $\left|a_{ii}-\lambda\right|=\left|b_{ii}\right|\leq\sum_{j\neq i}^{n}\left|b_{% ij}\right|=\sum_{j\neq i}^{n}\left|a_{ij}\right|$, which is Gerschgorin circle theorem. On the other hand, let’s assume Gerschgorin circle theorem is true, and let $A$ be a strictly diagonally dominant $n\times n$ complex matrix. Then, since the absolute value of each disc center $\left|a_{ii}\right|$ is strictly greater than the same disc radius $\sum_{j\neq i}^{n}\left|a_{ij}\right|$, the point $\lambda=0$ can’t belong to any circle, so it doesn’t belong to the spectrum of $A$, which therefore can’t be singular.

Title Levy-Desplanques theorem LevyDesplanquesTheorem 2013-03-22 15:34:50 2013-03-22 15:34:50 Andrea Ambrosio (7332) Andrea Ambrosio (7332) 9 Andrea Ambrosio (7332) Theorem msc 15-00