# Show that a tridiagonal matrix is nonsingular

How would you show that a given tridiagonal matrix is nonsingular?

I have a tridiagonal linear system and I would like to show it's nonsingular, which means zero cannot be an eigenvalue.

The book I am using suggests that one use the Gershgorin circle theorem

$$D_i = \{z:|z-a_{ii}|\} \le \sum_{j\ne i} a_{ij}$$

• Is the matrix also Toeplitz? Oct 30 '17 at 21:42

Do you have a specific triagonal matrix in mind? If so, just compute the Gersgorin Circles, and make sure they don't intersect the origin of the complex plane. If your matrix is $n \times n$ you have $n$ circles. The center of the $j$-th circle is the diagonal entry of the $j-$th column. The radius of the $j$-th circle is the sum of the absolute values of the off-diagonal entries of the $j$-th column. Since your matrix is triagonal, this will be extremely easy to compute. You could also define Gersgorin Circles in terms of the rows of your matrix as well.
EDIT: I made a slight correct, the radius of the $j$-th circles comes from adding the absolute values of the off-diagonal entries of the $j$-th column.