Critical points of matrix

In summary, critical points of a matrix refer to specific values or conditions that provide insights into the matrix's properties, such as eigenvalues, determinants, and stability. These points can indicate where the matrix undergoes significant changes, impacting solutions to linear equations, optimization problems, and system dynamics. Understanding critical points aids in analyzing the behavior of systems represented by matrices in various fields, including engineering, physics, and economics.
  • #1
TanWu
17
5
Homework Statement
Give their critical points of this system which can be written as a matrix:
##x^{\prime}=x - 2y, y^{\prime}=-2x+ 4y##
Relevant Equations
##x^{\prime}=x - 2y, y^{\prime}=-2x+ 4y##
My attempt is:

Condition for critical point is ##x' = y' = 0##,
##0 = x - 2y \implies 2y = x##
##-2x + dy = 0##
Then ##-4y + 4y = 0##

However, this means that critical points are ##(2y, y)## as system is linearly dependent (both equations are the same) where ##y \in \mathbb{R}##. However, that means there are infinitely many critical points which I have a doubt about.

I express gratitude to those who help.
 
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  • #2
TanWu said:
However, that means there are infinitely many critical points which I have a doubt about.
Why so?
 
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  • #3
TanWu said:
Homework Statement: Give their critical points of this system which can be written as a matrix:
##x^{\prime}=x - 2y, y^{\prime}=-2x+ 4y##
Relevant Equations: ##x^{\prime}=x - 2y, y^{\prime}=-2x+ 4y##

My attempt is:

Condition for critical point is ##x' = y' = 0##,
##0 = x - 2y \implies 2y = x##
##-2x + dy = 0##
Do you mean ##-2x + 4y = 0## here?
TanWu said:
Then ##-4y + 4y = 0##

However, this means that critical points are ##(2y, y)## as system is linearly dependent (both equations are the same) where ##y \in \mathbb{R}##. However, that means there are infinitely many critical points which I have a doubt about.
Your work looks ok to me. If you have doubts, have you checked that you got the initial problem equations right?
 
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  • #4
docnet said:
Why so?
FactChecker said:
Do you mean ##-2x + 4y = 0## here?

Your work looks ok to me. If you have doubts, have you checked that you got the initial problem equations right?
Thank you Sirs. I apoglize, that is a typo of me. Yes, got the initial problem equations correct. I was only expecting one critical point.
 
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  • #5
TanWu said:
Thank you Sirs. I apoglize, that is a typo of me.
Mathematics is very unforgiving in many ways. It's a learned skill to review your work very carefully.
TanWu said:
Yes, got the initial problem equations correct. I was only expecting one critical point.
You did a good job! The problems where you get a different answer than you expected are ones that really test you.
 
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FAQ: Critical points of matrix

What are critical points of a matrix?

Critical points of a matrix refer to the points in the matrix where certain properties, such as eigenvalues or determinants, exhibit significant changes. In the context of optimization, critical points can also refer to points where the gradient of a function defined by the matrix is zero, indicating potential maxima, minima, or saddle points.

How do you find the critical points of a matrix?

To find the critical points of a matrix, you typically need to compute the derivative of a function associated with the matrix (e.g., a quadratic form) and set it equal to zero. For eigenvalues, critical points can be found by solving the characteristic polynomial of the matrix, which is obtained from the determinant of the matrix minus a scalar times the identity matrix.

What role do eigenvalues play in determining critical points?

Eigenvalues are crucial in determining the stability and nature of critical points in a matrix. They provide information about the behavior of linear transformations represented by the matrix. If all eigenvalues are positive, the critical point is a local minimum; if all are negative, it is a local maximum; and if there are both positive and negative eigenvalues, the critical point is a saddle point.

Can critical points of a matrix be used in optimization problems?

Yes, critical points of a matrix are often used in optimization problems to identify local maxima, minima, or saddle points of multivariable functions. By analyzing the critical points, one can determine the optimal solutions and the nature of these solutions based on the second derivative test or the Hessian matrix.

What is the significance of the Hessian matrix in relation to critical points?

The Hessian matrix is the square matrix of second-order partial derivatives of a scalar-valued function. It plays a significant role in determining the nature of critical points. By evaluating the Hessian at a critical point, one can ascertain whether the point is a local minimum, local maximum, or saddle point based on the definiteness of the Hessian (positive definite, negative definite, or indefinite).

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