contestada

Suppose that v is an eigenvector of matrix A with eigenvalue λA, and it is also an eigenvector of matrix B with eigenvalue λB. (a) Show that v is an eigenvector of A + B and find its associated eigenvalue. (b) Show that v is an eigenvector of AB and find its associated eigenvalue.

Respuesta :

Answer:

(a) Yes, [tex]λ_{A}+λ_{B}[/tex]

(b) Yes, [tex]λ_{A}λ_{B}[/tex]

Step-by-step explanation:

First, lets understand what are eigenvectors and eigenvalues?

Note: I am using the notation [tex]λ_{A}[/tex] to denote Lambda(A) sign.

[tex]v[/tex] is an eigenvector of matrix A with eigenvalue [tex]λ_{A}[/tex]

[tex]v[/tex] is also eigenvector of matrix B with eigenvalue [tex]λ_{B}[/tex]

So we can write this in equation form as

[tex]Av=λ_{A}v[/tex]

So what does this equation say?

When you multiply any vector by A they do change their direction. any vector  that is in the same direction as of [tex]Av[/tex], then this [tex]v[/tex]  is called the eigenvector of [tex]A[/tex]. [tex] Av [/tex] is [tex]λ_{A}[/tex] times the original [tex]v[/tex]. The number [tex]λ_{A}[/tex] is the eigenvalue of A.

[tex]λ_{A}[/tex] this number is very important and tells us what is happening when we multiply [tex] Av [/tex]. Is it shrinking or expanding or reversed or something else?

It tells us everything we need to know!

Bonus:

By the way you can find out the eigenvalue of [tex] Av [/tex] by using the following equation:

[tex]det(A-λI)=0[/tex]

where I is identity matrix of the size of same as A.

Now lets come to the solution!

(a) Show that [tex]v[/tex] is an eigenvector of [tex]A + B[/tex] and find its associated eigenvalue.

The eigenvalues of [tex]A[/tex] and [tex]B[/tex] are [tex]λ_{A}[/tex] and [tex]λ_{B}[/tex], then

[tex](A+B)(v)=Av+Bv=(λ_{A})v + (λ_{B})v=(λ_{A}+λ_{B})(v)[/tex]

so,  [tex](A+B)(v)=(λ_{A}+λ_{B})(v)[/tex]

which means that [tex]v[/tex] is also an eigenvector of [tex]A+B[/tex] and the associated eigenvalues are [tex]λ_{A}+λ_{B}[/tex]

(b) Show that [tex]v[/tex] is an eigenvector of [tex]AB[/tex] and find its associated eigenvalue.

The eigenvalues of [tex]A[/tex] and [tex]B[/tex] are [tex]λ_{A}[/tex] and [tex]λ_{B}[/tex], then

[tex](AB)(v)=A(Bv)=A(λ_{B})=λ_{B}(Av)=λ_{B}λ_{A}(v)=λ_{A}λ_{B}(v)[/tex]

so,  

[tex](AB)(v)=λ_{A}λ_{B}(v)[/tex]

which means that [tex]v[/tex] is also an eigenvector of [tex]AB[/tex] and the associated eigenvalues are [tex]λ_{A}λ_{B}[/tex]