- #1
vish_maths
- 61
- 1
Let T be an operator on the vector space V and let λ1, ... , λn be it's eigen values including multiplicity .
Lets find the eigen values for the operator T*T then ( where T* refers to the adjoint operator . <u,v> denotes inner product of u and v )
< Tv , Tv > = < λv, λv >
= λλ° < v , v >
= |λ|2 < v , v>
( where λ° is the conjugate of λ . )
=> <T*T v, v> = < |λ|2 v, v >
=> T*T has an eigen value |λ|2 for the same eigen vector v which T possesses.
This can be inferred because :
Suppose f is a linear functional on V . Then there is a unique vector v in V such that f(u) =<u,v> for all u in V.
( A linear functional on V is a linear map from V to the scalars F , As sheldon axler pg 117 says it ) The above argument must imply then that : T*T v = |λ|2 v ( Doesn't this imply v is an eigen vector of T*T ? )
now, let's consider a matrix $$M(T) = \begin{bmatrix}
1 & 3 \\
0 & 2
\end{bmatrix}$$
[ 3 1 ]T is clearly an eigen vector with eigen value = 2 .
(T* T ) =\begin{bmatrix} 1 & 0 \\ 3 & 2 \end{bmatrix} multiplied by \begin{bmatrix}
1 & 3 \\
0 & 2
\end{bmatrix}$$
= \begin{bmatrix}
1 & 3 \\
3 & 13
\end{bmatrix}$$
M(T* T ) upon multiplication with [ 3 1 ]T should produce a vector equal to
4 [ 3 1 ]T
however , \begin{bmatrix}
1 & 3 \\
3 & 13
\end{bmatrix} multiplied by [ 3 1 ]T = [ 6 22 ]T .
Can you please advise why this paradox exists ? Am i making a mistake somewhere ?
Thanks
Lets find the eigen values for the operator T*T then ( where T* refers to the adjoint operator . <u,v> denotes inner product of u and v )
< Tv , Tv > = < λv, λv >
= λλ° < v , v >
= |λ|2 < v , v>
( where λ° is the conjugate of λ . )
=> <T*T v, v> = < |λ|2 v, v >
=> T*T has an eigen value |λ|2 for the same eigen vector v which T possesses.
This can be inferred because :
Suppose f is a linear functional on V . Then there is a unique vector v in V such that f(u) =<u,v> for all u in V.
( A linear functional on V is a linear map from V to the scalars F , As sheldon axler pg 117 says it ) The above argument must imply then that : T*T v = |λ|2 v ( Doesn't this imply v is an eigen vector of T*T ? )
now, let's consider a matrix $$M(T) = \begin{bmatrix}
1 & 3 \\
0 & 2
\end{bmatrix}$$
[ 3 1 ]T is clearly an eigen vector with eigen value = 2 .
(T* T ) =\begin{bmatrix} 1 & 0 \\ 3 & 2 \end{bmatrix} multiplied by \begin{bmatrix}
1 & 3 \\
0 & 2
\end{bmatrix}$$
= \begin{bmatrix}
1 & 3 \\
3 & 13
\end{bmatrix}$$
M(T* T ) upon multiplication with [ 3 1 ]T should produce a vector equal to
4 [ 3 1 ]T
however , \begin{bmatrix}
1 & 3 \\
3 & 13
\end{bmatrix} multiplied by [ 3 1 ]T = [ 6 22 ]T .
Can you please advise why this paradox exists ? Am i making a mistake somewhere ?
Thanks
Last edited: