Is the Matrix Symmetric Positive Definite for Cholesky Decomposition?

In summary, the conversation is about using the Cholesky method to solve a question involving a symmetric positive definite matrix. The person is unsure about how to proceed due to a possible error with the matrix. They are advised to go ahead and try to solve the question to see where the error lies.
  • #1
fonseh
529
2

Homework Statement


Here's the question :
1x1+ 2x2 +0x3 + 0x4 = 1
2x1+ 9x2 +1x3 + 0x4 = 6
0x1+ 1x2 +9x3 + 4x4 = 2
0x1+ 0x2 +4x3 + 3x4 = 8

I' m asked to solve this question using Choelsky method ( We need the symmetric positive definite matrix when we are using this method)

Homework Equations

The Attempt at a Solution


matrix A = $$\begin{bmatrix}
1 & 2& 0 & 0 \\
2 & 9 & 1 & 0 \\\
0 & 1 & 9 & 4 \\
0 & 0 & 4 & 3
\end {bmatrix} $$


the book stated that for the positive symmetric matrix , we need to ensure that max a_kj less than max a_ii ,
But , in this example , i found that the a_44 which is 3 is less than a_43 which is 4 ... So , how could this be symmetric positive definite matrix ?
How is it possible to solve using Choelsky method ?
 

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  • #2
fonseh said:
I' m asked to solve this question using Choelsky method
The guy is called Cholesky.

But it's clear what you are being asked, so go to work and see if and where it goes wrong !
 
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  • #3
question solved
 

FAQ: Is the Matrix Symmetric Positive Definite for Cholesky Decomposition?

What is symmetric positive definite?

Symmetric positive definite is a mathematical property of a square matrix. It means that the matrix is symmetric, meaning it is equal to its transpose, and all of its eigenvalues are positive.

Why is symmetric positive definite important?

Symmetric positive definite matrices have many useful properties and are commonly used in linear algebra and optimization. They guarantee that the matrix is invertible and can be used to solve systems of equations efficiently.

How do you determine if a matrix is symmetric positive definite?

To determine if a matrix is symmetric positive definite, you can check if it is symmetric and then calculate its eigenvalues. If all of the eigenvalues are positive, then the matrix is symmetric positive definite.

What is the significance of positive eigenvalues in symmetric positive definite matrices?

The positive eigenvalues in symmetric positive definite matrices indicate that the matrix is well-conditioned and has a strong correlation between its columns. This makes it useful for solving linear equations and performing other mathematical operations.

How is symmetric positive definite used in machine learning?

Symmetric positive definite matrices are commonly used in machine learning algorithms, such as principal component analysis and linear discriminant analysis. They help to reduce the dimensionality of data while preserving its important features.

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