And x2 = x2 (meaning that x2 is arbitrary).zak100 said:Homework Statement
Consider the following Matrix:
Row1 = 2 2
Row2 = 5 -1
Find its Eigen Vectors
Homework Equations
Ax = λx & det(A − λI)= 0.
The Attempt at a Solution
First find the det(A − λI)= 0. which gives a quadratic eq.
roots are
λ1 = -3 and λ2 = 4 (Eigen values)
Then using λ1, I got:
x1= -2/5 x2
zak100 said:Now how can i find out the vectors?
Some body please guide me.
Zulfi.
Who are you replying to? Use the Quote feature to copy what someone else is saying.zak100 said:Hi,
You are not correct.
That looks good. To summarize what you have, if one eigenvalue is ##\lambda = -3##, its associated eigenvector is ##\vec x = \begin{bmatrix} 2 \\ -5\end{bmatrix}##. You can and should verify this fact by confirming that ##A\begin{bmatrix} 2 \\ -5\end{bmatrix} = -3 \begin{bmatrix} 2 \\ -5\end{bmatrix}##, using the matrix you wrote in post #1.zak100 said:It says: -2 5. I don't know what would be the vector if:
x1 = -2x2.
Will it be : -2 1
It says about my question:
This means that, while there are infinitely many nonzero solutions (solution vectors) of the
equation Ax = −3x, they all satisfy the condition that the first entry x1 is −2/5 times the
second entry x2.
[ 2t ] = t [2 ]
−5t -5
(Note my square bracket is small. It should enclose 2t & -5t and the other one should enclose 2 -5)
,
where t is any real number. The nonzero vectors x that satisfy Ax = −3x are called
eigenvectors associated with the eigenvalue = −3. One such eigenvector is
u1 =[ 2 −5]
An eigenvector is a vector that does not change direction when a linear transformation is applied to it. In other words, when a matrix is multiplied by an eigenvector, the resulting vector is a scalar multiple of the original eigenvector.
To find the eigenvectors for a 2*2 matrix, you will need to find the eigenvalues first. You can do this by solving the characteristic equation, which is obtained by setting the determinant of the matrix minus the identity matrix equal to 0. Once you have the eigenvalues, you can plug them back into the original matrix to solve for the corresponding eigenvectors.
Eigenvectors are important in linear algebra because they provide a way to understand how a linear transformation affects a vector. They also help to simplify calculations, as it is often easier to work with eigenvectors than with the original matrix.
Yes, a matrix can have any number of eigenvectors. However, for a 2*2 matrix, there can be at most two linearly independent eigenvectors.
Eigenvectors have many real-life applications, such as in physics, engineering, and economics. They are used to study the behavior of complex systems, make predictions, and solve differential equations. For example, in physics, eigenvectors are used to find the normal modes of a vibrating system, and in economics, they are used to analyze the stability of a financial system.