For any  m X m  matrix A, the number of 0 eigenvalues is at least equal to  dim nullspace(A),

dim nullspace(A) <= number of 0 eigenvalues <= m

while the eigenvectors corresponding to those 0 eigenvalues belong to nullspace(A).

For diagonalizable matrix A, the number of 0 eigenvalues is precisely  dim nullspace(A)  while the corresponding eigenvectors span  nullspace(A).   The real and imaginary parts of the eigenvectors remaining span  range(A).

TRANSPOSE.

Likewise, for any  m X n  matrix A,
rank(A^T) + dim nullspace(A^T) = m

For any square  m X m  matrix A, the number of 0 eigenvalues is at least equal to  dim nullspace(A^T)=dim nullspace(A)  while the left-eigenvectors (eigenvectors of A^T) corresponding to those 0 eigenvalues belong to  nullspace(A^T).

For diagonalizable A, the number of 0 eigenvalues is precisely  dim nullspace(A^T)  while the corresponding left-eigenvectors
span  nullspace(A^T).  The real and imaginary parts of the left-eigenvectors remaining span  range(A^T).

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