Positive definitive matrix in wikipedia

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Discussion Overview

The discussion revolves around the interpretation of a mathematical expression found in the Wikipedia article on positive definite matrices. Participants are seeking clarification on how a specific expression is derived in the context of matrix multiplication and inner products.

Discussion Character

  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant questions the derivation of the expression 2x1^2 -2x1x2+2x2^2-2x2x3+2x3^2 from the example in the Wikipedia article.
  • Another participant explains that the expression results from multiplying a row vector by a column vector, which corresponds to an inner product.
  • A participant expresses confusion regarding the absence of certain variables (e.g., x3) in the expression, questioning how to interpret this in the context of the example.
  • A response clarifies that the absence of x3 in a specific term is due to a zero element in the corresponding matrix, indicating that the expression has been expanded correctly.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus, as there is ongoing confusion about the expression's derivation and the role of specific variables in the matrix.

Contextual Notes

There are unresolved questions regarding the assumptions made in the matrix representation and the implications of zero elements in the matrix affecting the resulting expressions.

de_student
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Hi all. I have a quick question

In this wikilink en.wikipedia.org/wiki/Positive-definite_matrix in the first example

I don't get how they get 2x1^2 -2x1x2+2x2^2-2x2x3+2x3^2 in the third line.

Can anyone bother to explain?

Thanks a lot
 
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de_student said:
Hi all. I have a quick question

In this wikilink en.wikipedia.org/wiki/Positive-definite_matrix in the first example

I don't get how they get 2x1^2 -2x1x2+2x2^2-2x2x3+2x3^2 in the third line.

Can anyone bother to explain?

Thanks a lot

Hey de_student and welcome to the forums.

For that line they are multiplying a row vector by a column vector which translates to an inner product. So if the row vector is [a, b, c] and the column vector is [x, y, z]^T then the result will be ax + by + cz using the standard definition of matrix multiplication.
 
Hi. Thanks for prompt answer. But my question is in the example there is not always [c]-member from row vector. like there is no x3 in the member (2x1-x2). How can I deal with that?
 
de_student said:
Hi. Thanks for prompt answer. But my question is in the example there is not always [c]-member from row vector. like there is no x3 in the member (2x1-x2). How can I deal with that?

I'm not sure what you mean, but the expression has been expanded out in the right manner. The reason there is no x3 for this term is because the M matrix has a 0 element at row 3, column 1. This is why there is no x3 element for this particular case in that particular cell.
 

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