Finding Eigenvalues of an Arbitrary Matrix

[himurakenshin]

How can i find the eigen value(s) of A - (alpha)I
where A is an arbitrary matrix ?

 

[HallsofIvy]

Your question is ambiguous. Do you mean just find the eigenvalues of A- which would mean solving the equation det(A- alpha*I)= 0 for alpha or do you mean specifically finding eigenvalues of A- alpha*I for a given value of I?

 

[himurakenshin]

sorry, where I is the identity matrix.
the matrix is C=(A-alpha*I)
I need to find the eigen values of C

 

[matt grime]

the eigenvalues of any square matrix, call it M, are the roots of the polynomial in x

det(M-xI)

although if you know the eigen values of A you know them of C too.

 

[himurakenshin]

the eigenvalues of any square matrix, call it M, are the roots of the polynomial in x

det(M-xI)

yes I know this, but I don't know how to find the eigen value of that paticular matrix (A can be any matrix). The actual question is that I have to prove that lambda is an eigen value of A only if (lamda - alpha) is an eigen value of C

 

[matt grime]

well, that wasn't waht you asked was it?

t is an eigenvalue of M if and only if M-tI is not invertible.

let a be alpha
If C-tI=A-aI-tI is not invertible, then A-(a+t)I is not invertible, can you fill in the blanks?

 

[himurakenshin]

got it. thanks a lot :)