(a) Write down an invertible matrix A whose Gershgorin domain contains 0. (b) Can you find an ex

step 1 So here we're given that A is invertible, and we have the equation that A times B is equal to A

(a) Write down an invertible matrix A whose Gershgorin...

Key Concepts

Strict Diagonal Dominance

A square matrix is said to be strictly diagonally dominant if, for each row, the magnitude of the diagonal entry is greater than the sum of the magnitudes of all the other entries in that row. This condition not only ensures the invertibility of the matrix but also often guarantees favorable numerical stability properties. In contexts where one is asked for matrices that are both strictly diagonally dominant and have certain eigenvalue characteristics (as described by the Gershgorin theorem), this property helps in maintaining the desired balance between the matrix entries while still fulfilling the eigenvalue location criteria.

Invertible Matrix

An invertible matrix is one that has a non-zero determinant, which guarantees the existence of an inverse matrix. This property is central in linear algebra because it implies that the matrix corresponds to a one-to-one linear transformation, ensuring that each input is uniquely mapped to an output. In problem settings, constructing an invertible matrix often involves careful selection of entries to avoid singularity, even if additional conditions are imposed on other features such as the Gershgorin domain.

Gershgorin Circle Theorem

The Gershgorin Circle Theorem provides a way to estimate the locations of eigenvalues of a square matrix by considering discs (or regions) in the complex plane. Each disc is centered at a diagonal entry of the matrix, with a radius equal to the sum of the absolute values of the non-diagonal entries in that row. This theorem is useful for demonstrating the properties of matrices, such as invertibility or the potential for eigenvalues to lie within a specified region, even if the matrix does not appear obviously well-behaved.

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