Eigenvalue vector x

1. Dec 8, 2013

Miike012

Question one: in regards to two segments underlined in blue.

If (a,x) is an eigenvalue and vector of A, that means

Ax = ax, where a is a real number.

My question is, is Amx = amx, where m in an integer greater than 1?

Question 2: in regards to two segments underlined in red.

I know that I*x = x, where I is the nxn identity matrix and x is a vector in Rn.
But the last part where the vector x is factored, shouldn't the 1 be an I?

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2. Dec 8, 2013

Dick

Yes, $A^mx=a^mx$. You should try to prove that if you aren't sure. For the second question what they factored out are scalars. Since Ix=1x what you factor out should be the scalar 1, not the matrix I.

3. Dec 8, 2013

Ray Vickson

Note: do NOT assume that the eigenvalue is real: it may not be. Some matrices have only real eigenvalues, others have some that are complex.