- #1

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If A is

**n x n**and r is not 0.

Show that

**CrA(x) = (r^n) * CA(x/r)**

What rule should I think of in defanition.

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- Thread starter John Smith
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In summary, an eigenvalue in linear algebra is a scalar value that represents the scaling factor of an eigenvector. It is calculated by solving the characteristic equation det(A - λI) = 0 and is significant because it can provide information about the behavior of a system. Eigenvalues are closely related to eigenvectors, with the eigenvalue representing the scaling factor for the eigenvector. Real-world applications of eigenvalues include image and signal processing, machine learning, and data compression in various fields such as physics, economics, and engineering. They are also used to study quantum systems and analyze market trends. Overall, eigenvalues are a powerful tool for analyzing systems and making predictions.

- #1

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- 0

If A is

Show that

What rule should I think of in defanition.

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- #2

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Its okey I did find it out.

- #3

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In linear algebra, an eigenvalue is a scalar value that represents the scaling factor of an eigenvector. It is a special characteristic of a square matrix and is often used to simplify calculations and make predictions about the behavior of a system.

The eigenvalues of a matrix can be calculated by solving the characteristic equation det(A - λI) = 0, where A is the matrix, λ is the eigenvalue, and I is the identity matrix. This equation yields a set of values for λ, which are the eigenvalues of the matrix.

Eigenvalues are important in linear algebra because they can provide information about the behavior of a system. They can be used to determine stability, convergence, and other properties of a matrix. They also play a key role in diagonalizing matrices and solving differential equations.

Eigenvalues and eigenvectors are closely related in linear algebra. An eigenvector is a vector that is scaled by its corresponding eigenvalue when multiplied by a matrix. In other words, the eigenvalue represents the scaling factor for the eigenvector. Additionally, the number of eigenvalues of a matrix is equal to the number of eigenvectors.

Eigenvalues and eigenvectors have numerous applications in various fields, including physics, economics, and engineering. They are used in image and signal processing, machine learning, and data compression, among others. In physics, eigenvalues are used to study the behavior of quantum systems, while in economics, they are used in analyzing market trends and forecasting. In general, eigenvalues provide a powerful tool for analyzing systems and making predictions about their behavior.

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