- #1
arpon
- 235
- 16
Suppose, ##A## is an idempotent matrix, i.e, ##A^2=A##.
For idempotent matrix, the eigenvalues are ##1## and ##0##.
Here, the eigenspace corresponding to eigenvalue ##1## is the column space, and the eigenspace corresponding to eigenvalue ##0## is the null space.
But eigenspaces for distinct eigenvalues of a matrix have intersection ##\{0\}##.
So, null space and column space are complementary for idempotent matrix. That means the row space and column space are the same for idempotent matrix.
Is this argument correct?
For idempotent matrix, the eigenvalues are ##1## and ##0##.
Here, the eigenspace corresponding to eigenvalue ##1## is the column space, and the eigenspace corresponding to eigenvalue ##0## is the null space.
But eigenspaces for distinct eigenvalues of a matrix have intersection ##\{0\}##.
So, null space and column space are complementary for idempotent matrix. That means the row space and column space are the same for idempotent matrix.
Is this argument correct?