Inverse matrices with scalar multiplication.

[brydustin]

I was surprised that I have never had to do this in so long and forgot the basic way to factor out a scalar multiple when a matrix is raised to a certain power (for example -1 for inverse matrices).

Basically, I just want some confirmation:

(λT)^n= λ^n (T^n ) ∶ for λ ϵ F and Tϵ L(V).
For example:
(λT)^(-1)=(1/λ) T^(-1)
or…. (λI-T)^(-1)=(1/λ) (I-T/λ)^(-1)

 

[Outlined]

we have tex, sup and sub tags. Request to use them.

 

[Landau]

Basically, I just want some confirmation:

(λT)^n= λ^n (T^n ) ∶ for λ ϵ F and Tϵ L(V).

Remember that matrix multiplication is just composition of linear maps.

For all x we have

$$((\lambda T)\circ (\lambda T))(x)=(\lambda T)(\lambda (Tx))=\lambda(\lambda T(Tx))=\lambda^2(T\circ T(x))=(\lambda^2 T^2) x$$

using linearity. So it holds for n=2. By induction it holds for all n.

 

[brydustin]

Remember that matrix multiplication is just composition of linear maps.

For all x we have

$$((\lambda T)\circ (\lambda T))(x)=(\lambda T)(\lambda (Tx))=\lambda(\lambda T(Tx))=\lambda^2(T\circ T(x))=(\lambda^2 T^2) x$$

using linearity. So it holds for n=2. By induction it holds for all n.

great! I thank this wraps up this thread then... I assume then that it holds for strictly negative values of n as well...

 

[Landau]

Yes, if T is invertible and lambda =/0 then

$$(\lambda^{-1}T^{-1}\circ \lambda T)(x)=(\lambda^{-1}T^{-1})(\lambda Tx)=T^{-1}Tx=x$$

hence

$$(\lambda T)^{-1}=\lambda^{-1}T^{-1}.$$