Eigenvector Question: Is v=(2,1) the same as v=(4,2)?

• barksdalemc
In summary, eigenvectors are not equivalent if they have different magnitudes, they can have negative components, they are calculated by finding the solution to the characteristic equation of a matrix, a matrix can have multiple eigenvectors, and they are important in linear algebra for various applications.
barksdalemc
If I find a given eigenvector , that vector spans the entire eigenspace defined by that eigenvalue correct? Let's say I get v=(2,1) as an eigenvector. That is the same as saying v=(4,2) right? since they are spanning the same space?

yes that is correct

thanks a lot

1. Are the two eigenvectors, v=(2,1) and v=(4,2), equivalent?

No, the two eigenvectors are not equivalent. Although they have the same direction, their magnitudes are different. Eigenvectors are only considered equivalent if they have the same direction and magnitude.

2. Can an eigenvector have negative components?

Yes, an eigenvector can have negative components. The components of an eigenvector represent the scaling factors for each corresponding basis vector, and these scaling factors can be positive or negative.

3. How is an eigenvector calculated?

An eigenvector is calculated by finding the solution to the characteristic equation of a square matrix. The characteristic equation is (A - λI)x = 0, where A is the matrix, λ is the eigenvalue, and x is the eigenvector.

4. Can a matrix have more than one eigenvector?

Yes, a matrix can have multiple eigenvectors. In fact, it is common for a matrix to have multiple eigenvectors, each corresponding to a different eigenvalue. The number of eigenvectors a matrix has is equal to its dimension.

5. What is the significance of eigenvectors in linear algebra?

Eigenvectors are important in linear algebra because they represent the directions along which a linear transformation only results in a scaling of the vector. This makes them useful for solving systems of linear equations, finding stable equilibria in dynamical systems, and performing dimensionality reduction in data analysis.

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