That's a decent start. Next, show that A(Sx) = [itex]\lambda[/itex]x. That's what it means to say that Sx is an eigenvector of A corresponding to [itex]\lambda[/itex].WTFsandwich said:Homework Statement
Let B = S^-1 * A * S and x be an eigenvector of B belonging to an eigenvalue [tex]\lambda[/tex]. Show S*x is an eigenvector of A belonging to [tex]\lambda[/tex].
Homework Equations
The Attempt at a Solution
The only place I can think of to start, is that B*x = [tex]\lambda[/tex]*x.
However, even starting with that, I can't figure out where to go next.
Could someone point me in the right direction?
Slight correction: You want to show that A(Sx) = [itex]\lambda[/itex](Sx)Mark44 said:That's a decent start. Next, show that A(Sx) = [itex]\lambda[/itex]x. That's what it means to say that Sx is an eigenvector of A corresponding to [itex]\lambda[/itex].
Multiply by x now.WTFsandwich said:What do I use to show that?
The only new information I've got that might be helpful is that A = S * B * S^-1
Multiplying on the left by S gives A*S = S*B
After doing that, I'm stuck again. I feel like this is the right track, but I don't know how to relate this back to what I'm trying to prove.
An eigenvector is a vector that, when multiplied by a square matrix, results in a scalar multiple of itself. In other words, the direction of the vector remains unchanged but its magnitude is scaled by a constant factor.
Eigenvectors are significant because they represent the directions along which a linear transformation has the simplest effect. They are used in many applications, including computer graphics, data analysis, and quantum mechanics.
Eigenvectors and eigenvalues are closely related. An eigenvalue is the scalar multiple by which an eigenvector is scaled when multiplied by a matrix. In other words, the eigenvalue represents the magnitude of the transformation along the corresponding eigenvector.
Eigenvectors and eigenvalues are calculated by finding the roots of the characteristic polynomial of a square matrix. This polynomial is obtained by subtracting the identity matrix multiplied by the eigenvalue from the original matrix, and then finding the determinant of this resulting matrix.
Understanding eigenvectors is important in many fields, including physics, engineering, and computer science. They are used in image and signal processing, data compression, and machine learning algorithms. In physics, eigenvectors play a crucial role in understanding quantum mechanics and the behavior of particles in a quantum system.