Eigenvalues and matrix rank, any relation?

Question

Is it true to say that matrix rank is equal to the number of eigenvalues different from 0?

If not please give me contradiction example since I didn't find one.

Answer 1

Consider $$ \begin {pmatrix} 0&1\\0&0\\ \end{pmatrix}$$.

This matrix has both Eigen values zero but its rank is one.

Answer 2

The dimension of eigenspace corresponding to $\lambda=0$ is the nullspace of $A$ therefore, by rank-nullity theorem, the statement is true only when the geometric and algebraic multiplicity are equal for $\lambda=0$, that is when the eigenvalue $\lambda=0$ is not defective.

Refer to the related

• Finding the rank of the matrix directly from eigenvalues
• Relation between rank and number of distinct eigenvalues of a matrix