Eigenvalue Vector x: Question 1 and 2

[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 A^mx = a^mx, 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?

 

[Dick]

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 A^mx = a^mx, 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?

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.

 

[Ray Vickson]

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 A^mx = a^mx, 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?

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.