Which means that the eigenvalues are ±√6.SMA_01 said:Homework Statement
I'm supposed to find all the eigenvalues for this 2x2 matrix:
0 2
3 0
Homework Equations
The Attempt at a Solution
When I tried to do it with no row interchanges, I got the characteristic characteristic equation:
λ2-6=0
The latter part is wrong.SMA_01 said:So, instead of solving this, I interchanged the rows of my matrix to get:
3 0
0 2
And then my determinant would be negative right? I got λ=2,3...I was wondering if I did this right?
Eigenvalues are a mathematical concept used in linear algebra to describe the behavior of a transformation or a matrix. They represent the values that remain unchanged when an operation is applied to a vector or matrix.
The calculation of eigenvalues involves finding the roots of the characteristic polynomial of a given matrix. This can be done using various techniques, such as the QR algorithm or the power method.
Eigenvalues have various applications in fields such as physics, engineering, and computer science. They are used to solve differential equations, analyze the stability of systems, and compress data in machine learning algorithms, among others.
Eigenvalues and eigenvectors are closely related concepts. Eigenvectors are the corresponding vectors to eigenvalues, and they represent the directions along which a transformation or matrix has a constant scale factor. Eigenvalues and eigenvectors are used together to solve systems of linear equations and diagonalize matrices.
Yes, eigenvalues can be visualized in a geometric way. For example, in two-dimensional space, eigenvalues can be represented as the lengths of the semi-axes of an ellipse, where the eigenvectors are the directions of the semi-axes. In three-dimensional space, eigenvalues can be represented as the lengths of the semi-axes of an ellipsoid.