In very brief comment express please write it by using keyb

In very brief comment express, ( please write it by using keyboard)

What does it mean to say that lambda is an eigenvalue of a matrix A?

What does it mean to say that vector x is an eigenvector corresponding to lambda?

Solution

To understand What does it mean eigenvalues/eigenvectors, you must first understand what is matrices and vectors.

In number of situations, the objects you study and the stuff you can do with them relate to vectors and linear transformations, which are represented as matrices.

So, in many many interesting situations, important relations are expressed as

y =Mx

where y and x are vectors and M is a matrix. This ranges from systems of linear equations you have to solve (which occurs virtually everywhere in science and engineering) to more sophisticated engineering problems (finite element simulations). It also is the foundation for (a lot of) quantum mechanics.

y we can write as x where is scalar and x is the vector, means i.e when I apply M to x which is the eigenvector, I get another vector whose direction is the same as x but whose magnitude is scaled by some scalar which we call eigenvalue .

So why are these eigenvectors and eigenvalues important? Consider the eigenvector corresponding to the maximum (absolute) eigenvalue. If we take a vector along this eigenvector, then the action of the matrix is maximum. No other vector when acted by this matrix will get stretched as much as this eigenvector.