Physical significance of eigen vectors of Covariance matrix

In summary, the conversation discusses a doubt regarding the physical significance of eigen vectors of the covariance matrix and how they are used as principal components for dimensionality reduction. The definition of principal components is mentioned and it is pointed out that it cannot be proven. The discussion then moves on to how researchers arrive at definitions through observations, using the example of finding orthogonal eigen vectors in a symmetrical covariance matrix. The speaker expresses interest in learning more about these processes without access to professors.
  • #1
dexterdev
194
1
Hi all,

I have a doubt regarding the physical significance of eigen vectors of the covariance matrix. I came to know that eigen vectors of covariance matrix are the principal components for dimensionality reduction etc, but how to prove it?
 
Physics news on Phys.org
  • #2
That's the definition of principal components. How can you prove a definition?
 
  • #3
OK Sir, I was not knowing that it was a definition. Let me ask one thing...How researchers arrive at definitions. ie here when ever you find eigen vectors of covariance matrix (which is symmetrical matrix) you find them orthogonal and regression scatter plot lies of the significant PCA1 and least significant PCA2 has less strength points (noise etc). Is definitions formed from observation. Just to know these sort of things since I have no access to any professors.
 

Related to Physical significance of eigen vectors of Covariance matrix

1. What is a covariance matrix and why is it important?

A covariance matrix is a square matrix that summarizes the relationship between multiple variables in a dataset. It is important because it allows us to understand how different variables are related to each other and to identify patterns and trends in the data.

2. What are eigen vectors and why are they significant in the context of a covariance matrix?

Eigen vectors are special vectors that represent the directions of maximum variability in a dataset. In the context of a covariance matrix, eigen vectors are significant because they help us understand the underlying structure and relationships between variables in the data.

3. How do eigen vectors of a covariance matrix relate to principal component analysis (PCA)?

PCA is a statistical technique that uses the eigen vectors of a covariance matrix to identify the most important patterns and relationships in a dataset. The eigen vectors of the covariance matrix are used to create new variables, called principal components, which are linear combinations of the original variables.

4. What is the physical significance of eigen vectors of a covariance matrix in real-world applications?

The physical significance of eigen vectors of a covariance matrix depends on the specific application. In fields such as physics and engineering, eigen vectors can represent physical quantities such as forces or directions of motion. In data analysis, eigen vectors can represent important patterns or relationships in the data.

5. How can understanding the physical significance of eigen vectors of a covariance matrix help in data analysis and interpretation?

Understanding the physical significance of eigen vectors of a covariance matrix can help in data analysis and interpretation by providing insights into the underlying structure and relationships in the data. This can aid in identifying important variables, reducing dimensionality, and making predictions or classifications based on the data.

Similar threads

I How to Get Final Fisher Matrix from 2 Matrices
    • Set Theory, Logic, Probability, Statistics
Replies
11
Views
2K
I Changing diagonal elements of a matrix
    • Set Theory, Logic, Probability, Statistics
Replies
6
Views
1K
A Computing the Fisher Matrix numerically
    • Set Theory, Logic, Probability, Statistics
Replies
1
Views
1K
I Fisher matrix - equivalence or not between sequences
    • Set Theory, Logic, Probability, Statistics
Replies
3
Views
2K
I Vector visualization of multicollinearity
    • Set Theory, Logic, Probability, Statistics
Replies
1
Views
751
I Bayesian Stats: Resources about Mercer's Theorem for Gaussian Processes
    • Set Theory, Logic, Probability, Statistics
Replies
1
Views
725
A Covariance matrix for transformed variables
    • Set Theory, Logic, Probability, Statistics
Replies
5
Views
2K
I Weird covariance matrix for linear fit
    • Set Theory, Logic, Probability, Statistics
Replies
1
Views
870
A How to combine correlated vector-valued estimates
    • Set Theory, Logic, Probability, Statistics
Replies
11
Views
2K
A Contravariance, Covariance, stress and position vector
    • Classical Physics
Replies
24
Views
1K

Hot Threads

Recent Insights

Back
Top