What are the Eigenvalues and Eigenvectors of a Matrix?

Thread starter Jocey13
Start date Mar 13, 2008
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In summary, Eigenvalues and Eigenvectors are mathematical concepts used to understand the properties of a matrix. Eigenvalues are the scalar values that represent the scaling factor of the Eigenvectors, which are the special set of vectors that do not change direction after being multiplied by the matrix. These values and vectors play a crucial role in various applications, such as solving systems of linear equations, analyzing the stability of dynamic systems, and analyzing data in statistics and machine learning. Additionally, the Eigenvalues and Eigenvectors of a matrix can provide insights into its geometric and algebraic properties, making them important tools in linear algebra.

Mar 13, 2008

Jocey13

Homework Statement

Let: a matrix be: -5 -0.5
-0 -8

Find an invertible P and a diagonal D such that PDP(inverse)

Homework Equations

DET( (I)Lamda-A))= 0 for Eigenvalues

The Attempt at a Solution

when y=0 at the end matrix for finding the eigenvectors how do i make that a P

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Mar 13, 2008

ircdan

what are your eigenvalues? P is a matrix whose columns will be the eigenvectors of A

1. What is the parametric form question?

The parametric form question is a way of representing a mathematical equation or function using parameters. It involves expressing the variables in terms of another variable, usually represented as t, and then using a set of parametric equations to describe the relationship between the variables.

2. What are the advantages of using parametric form?

Parametric form allows for a more flexible and intuitive representation of equations or functions. It can be useful for solving complex equations, graphing curves, and analyzing the behavior of a system over time. Additionally, it can help to simplify calculations and make it easier to express the relationship between variables.

3. How do you convert an equation to parametric form?

To convert an equation to parametric form, you first need to identify the variables and express them in terms of a parameter, usually represented as t. Then, you can use a set of parametric equations to describe the relationship between the variables, often by using trigonometric functions. Finally, you can eliminate the parameter and simplify the equations to get the final parametric form.

4. What are some common applications of parametric form?

Parametric form is commonly used in physics, engineering, and computer graphics to describe the motion of objects, such as projectile motion or motion of planets. It is also used in parametric equations for curves, such as circles, ellipses, and parabolas. Additionally, it can be applied in optimization problems and in parametric statistics.

5. What are some potential limitations of using parametric form?

While parametric form can be useful in many applications, it may not be the most practical or efficient way of representing equations or functions in some cases. It can be more complex and difficult to understand, and may require additional calculations to convert back to standard form. Additionally, it may not always be possible to express a given equation in parametric form.