Column space of positive semidefinite matrix

[td21]

how to prove that
$$R(A)=\text{sum of} N(A-\lambda I)$$?

$\lambda$ is nonzero eignevalues of A

 

[td21]

well i can show that
1)if y$\in$R(A), then y=Ax=λx.
2)As (A−λI)x=0, x$\in$N(A−λI).

But how can we show that y$\in$ sum of N(A−λI)?

thanks

 

[td21]

Does this mean y= sum of eignevectors of A?

 

[td21]

anyone?thanks

 

[micromass]

Hi td21!

What about the zero matrix? This is positive semidefinite, and

$$R(0)=0~\text{and}~N(0-\lambda I)=N(0)=\text{entire space}$$