Eigen values of a matrix with a few diagonal rows.

Question

I have a square matrix $A$ of size 1000 x 1000. Some of the rows (not all) of this square matrix have only unit diagonal entries, i.e., for some rows $A_{ij} = 1$ when $i = j$ and rest of the elements of these rows are zero. I computed the eigenvalues of $A$ with Slepc and found that there are unit eigenvalues equal to the number of such rows (rows with the unit diagonal element). Is it normal? If yes, can you point me how to prove this? Or point to a suitable reference.

PS: I got matrix $A$ from finite element analysis while setting Dirichlet boundary conditions with the substitution technique.

Answer 1

Let $i_1<i_2<\ldots<i_k$ be the indices of the special rows. It follows that the matrix $A-I$ has all zeros on those $k$ rows. Consequently the rowspace $r(A-I)$ has dimension $\le 1000-k$. The orthogonal complement $r(A-I)^\perp$ therefore has dimension $\ge1000-(1000-k)=k$ (with equality iff the non-zero rows of $A-I$ are linearly independent). When viewed as column vectors, the vectors in $r(A-I)^\perp$ are eigenvectors of $A$ belonging to the eigenvalue $\lambda=1$. Thus the multiplicity of $\lambda=1$ as an eigenvalue of $A$ is at least $k$.

Answer 2

Let $i_1,\cdots,i_k$ be the indices of your special rows. Then consider $A^T$. $e_{i_1},\cdots,e_{i_k}$ are linearly independent eigenvectors of $A^T$ with associated eigenvalue $1$. Then $A^T$ admits the eigenvalue $1$ with multiplicity at least $k$, and $A$ too. $\square$