I have a general question regarding eigenvalues/vectors. Say you are

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In summary, eigenvalues and eigenvectors are mathematical concepts used in linear algebra to describe the behavior of linear transformations on vector spaces. They are calculated by solving a system of linear equations, specifically the characteristic equation for a given matrix. These values have various applications in mathematics, physics, and engineering, and can also have complex values. In data analysis, eigenvalues and eigenvectors can be used to simplify calculations, identify patterns, and reduce dimensionality through techniques like PCA and SVD.
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kr0z3n
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I have a general question regarding eigenvalues/vectors. Say you are working on a matrix
[ 1, 1; 4, 1] and you find the eigenvalues to be 3 and -1.
Does it matter which eigenvalue is first...meaning does it matter when it is λ1= 3 and λ2=-1 or the other way around?
When you write out the general solution in the form of c1*e^-t[1, -2] + c2*e^3t[1, 2] , how do you know which eigenvalue is written first? In other words why can the general solution not be written as c1*e^3t[1, 2] + c2*e^-t[1, -2]?
 
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No. It doesn't matter at all.
 
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Thank you
 

FAQ: I have a general question regarding eigenvalues/vectors. Say you are

What are eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are mathematical concepts used in linear algebra to describe the behavior of a linear transformation on a vector space. Eigenvalues represent the scalar values by which the eigenvectors are scaled when the linear transformation is applied.

How are eigenvalues and eigenvectors calculated?

Eigenvalues and eigenvectors are calculated by solving a system of linear equations, specifically the characteristic equation for a given matrix. This involves finding the roots of the characteristic polynomial, which is formed by subtracting the variable λ from the main diagonal of the matrix and taking the determinant. The corresponding eigenvectors can then be found by solving a system of equations using the eigenvalues.

What is the significance of eigenvalues and eigenvectors?

Eigenvalues and eigenvectors have various applications in mathematics, physics, and engineering. They are used to simplify complex calculations and analyze the behavior of linear systems. They also play a crucial role in the development of algorithms for data analysis and machine learning.

Can eigenvalues and eigenvectors have complex values?

Yes, eigenvalues and eigenvectors can have complex values. In fact, complex eigenvalues and eigenvectors are often encountered in real-world applications, especially in quantum mechanics and signal processing.

How can eigenvalues and eigenvectors be used in data analysis?

Eigenvalues and eigenvectors can be used in data analysis to reduce the dimensionality of a dataset and identify patterns or underlying structures. This is commonly done through techniques such as principal component analysis (PCA) and singular value decomposition (SVD).

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