Eigen values and cubic roots question

• iamsmooth
In summary, the speaker is discussing their difficulty in finding the roots of a cubic equation, specifically in relation to eigenvalues. They mention different options for solving cubic equations, such as factoring, guessing an answer, or using a formula.
iamsmooth
So I found the characteristic equation of a matrix, and I know the roots of the equation are supposed to be the eigenvalues. However, my equation is:

$$\lambda^3-2\lambda^2$$

I have double checked different row expansions to make sure this answer is correct. So don't worry about how I came to get that equation.
I'm just not sure how to get roots from this. Would it be:

$$\lambda^2(\lambda-2)$$

So that the roots are 0 and 2?

Basically I have trouble with cubic roots, I guess this is less of a question about eigenvalues than it is about cubic roots.

Thanks.

You are right about your example (note 0 is a double root).
In general cubic equations are no fun. Your options are:
1. Factor, like you did above
2. Guess an answer $\lambda_0$, then divide out $(\lambda-\lambda_0)$ and solve the resulting quadratic equation. This works well in constructed problems where you can easily see that a value like 0, 1, -1, 2 or -2 satisfies the equation.
3. Use the analog of the quadratic formula ($(-b\pm\sqrt{b^2-4ac})/2a$) for cubic equations. However it is messy, and unlike the quadratic formula hardy anyone knows it by heart.

1. What are eigenvalues and why are they important?

Eigenvalues are a concept in linear algebra that represent the scalar values by which a matrix multiplies a vector. They are important because they help us understand the behavior of a system or object by identifying its dominant features or characteristics.

2. How do you calculate eigenvalues and eigenvectors?

To calculate eigenvalues and eigenvectors, you need to find the characteristic polynomial of a matrix, which is obtained by subtracting the scalar variable from the main diagonal elements of the matrix and taking the determinant. Then, solve for the values of the scalar variable that make the determinant equal to zero. These values are the eigenvalues. To find the corresponding eigenvectors, plug in each eigenvalue into the characteristic polynomial and solve for the variables.

3. What is the relationship between eigenvalues and diagonalization?

Diagonalization is the process of transforming a matrix into a diagonal matrix by finding a matrix with the eigenvectors as its columns. The diagonal elements of this matrix are the eigenvalues. Therefore, the eigenvalues and eigenvectors play a crucial role in diagonalization.

4. Can a matrix have complex eigenvalues and eigenvectors?

Yes, a matrix can have complex eigenvalues and eigenvectors if the matrix itself contains complex numbers. This is because the characteristic polynomial of a matrix with complex numbers can have complex roots, which correspond to the eigenvalues, and the eigenvectors can also have complex components.

5. What is the significance of eigenvalues in data analysis and machine learning?

In data analysis and machine learning, eigenvalues play a crucial role in dimensionality reduction. By finding the eigenvalues and eigenvectors of a dataset or a correlation matrix, we can identify the most important features or variables that contribute the most to the variance in the data. These features can then be used to reduce the dimensionality of the data without losing much information, making data analysis and machine learning more efficient.

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