Positive definitive matrix in wikipedia

In summary, the conversation is about understanding the calculation of a positive-definite matrix in the first example provided in a wikipedia article. The question is about how the third line of the calculation, which includes a row vector and column vector, is derived. The response explains that the calculation is a result of multiplying a row vector by a column vector, which is equivalent to an inner product. The questioner then asks about the absence of a third element in the expression (2x1-x2) and the responder clarifies that the element is missing because it corresponds to a 0 element in the matrix.
  • #1
de_student
2
0
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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  • #2
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.
 
  • #3
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?
 
  • #4
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.
 
  • #5


Hi there,

I am happy to provide an explanation for the example in the Wikipedia article on positive definite matrices.

First, let's define what a positive definite matrix is. A positive definite matrix is a square matrix where all of its eigenvalues (the numbers that, when multiplied by the matrix, result in the same vector) are greater than 0. In simpler terms, this means that all the numbers in the matrix are positive and the matrix has certain mathematical properties that make it useful in various fields, such as optimization and statistics.

Now, let's take a closer look at the example in the Wikipedia article. The matrix in question is a 3x3 matrix, which means it has 3 rows and 3 columns. The third line in the example is showing how to calculate the determinant of this matrix, which is a value that can be used to determine if a matrix is positive definite.

The formula for calculating the determinant of a 3x3 matrix is:
det(A) = a11(a22a33 - a23a32) - a12(a21a33 - a23a31) + a13(a21a32 - a22a31)

In the example, the entries of the matrix are:
a11 = 2x1, a12 = -x1x2, a13 = 0
a21 = -x1x2, a22 = 2x2, a23 = -x2x3
a31 = 0, a32 = -x2x3, a33 = 2x3

Substituting these values into the formula, we get:
det(A) = 2x1(2x2x2x3 - (-x2x3)(-x2)) - (-x1x2)(-x1x3 - (-x2x3)(0)) + 0(2x1(-x2x3) - 2x2(-x1x3))

Simplifying this equation, we get:
det(A) = 4x1x2x2x3 + x1x2x2 - x1x2x2x2 - x1x2x2x3

Notice that the terms in the third line of the matrix correspond to the terms in the above equation. This is how they arrive at the expression 2x1^2 - 2
 

1. What is a positive definite matrix?

A positive definite matrix is a square matrix where all of the eigenvalues are positive. In other words, it is a matrix that has only positive diagonal elements and all of its principal minors (determinants of submatrices) are also positive.

2. What are the properties of a positive definite matrix?

A positive definite matrix has several important properties, including being symmetric (equal to its transpose), having all positive eigenvalues, and being invertible. It also has a unique Cholesky decomposition, which is useful for solving linear systems of equations.

3. How is a positive definite matrix used in mathematics and statistics?

A positive definite matrix is used in a variety of mathematical and statistical applications, including optimization problems, multivariate analysis, and machine learning algorithms. Its properties make it useful for solving systems of equations, minimizing functions, and characterizing the shape of a probability distribution.

4. How can a positive definite matrix be identified?

There are several ways to identify a positive definite matrix, including calculating its eigenvalues and checking that they are all positive, or using the Sylvester's criterion which states that a matrix is positive definite if and only if all of its leading principal minors are positive. Additionally, some software packages have built-in functions for checking the positive definiteness of a matrix.

5. Can a positive definite matrix have negative eigenvalues?

No, a positive definite matrix by definition has only positive eigenvalues. If a matrix has any negative eigenvalues, it cannot be positive definite. However, a matrix can have zero eigenvalues and still be positive definite, as long as all of its other eigenvalues are positive.

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