# Can the null space of matrix B be used for data projection?

• Sue_2010
In summary, the conversation discusses the concept of null space projection in order to project data to a direction that maximally preserves the Euclidean distance between two vectors. The speaker proposes using the row space of (x-y)' as the desired direction, but acknowledges that directly calculating the row space projection matrix may be difficult. Instead, the speaker suggests using an approximation method by finding the null space of a matrix B and projecting the data onto it. They also mention that for normalized vectors, the null space of B may be equal to the row space of A. The speaker asks for input and feedback on their approach.
Sue_2010
Hello everyone,

If I have a collection of data points (vectors), and x and y are two vectors among them. I want to project the data to a direction that the Euclidean distance between x and y is Maximally preserved. Then this direction should be the row space of (x-y)’, denoted as row( (x-y)’ ), right? (suppose column vectors.)

Now, suppose I have n pairs of such data points grouped as a matrix A, say

A = [
(x1 – y1)’
(x2 – y2)’

(xn –yn)’
]

For my problem, directly calculate the row space projection matrix P1 = A’*inv(AA’)*A is difficult, so I want to do some approximation.

Suppose all the vectors have been normalized, that is norm(x1) = norm(x2) = … = norm(xn) = norm(y1) = .. = norm(yn). I define matrix B as

B = [
(x1 + y1)’
(x2 + y2)’

(xn + yn)’
]

And plan to find the null space of B, denoted as null(B). It is easy for me to calculate the null space projection matrix P2 = I – B’*inv(BB’)*B. Note that, (xi – yi)’*(xi+yi) = 0 for normalized vectors.

I feel that null(B) = row(A) . Is it true? Or what’s the relationship between null(B) and row(A)? Can I make the conclusion that, if I project data to null(B), those pair of points (xi, yi) will also be maximally separated?

I’m waiting online. Any input will be appreciated! You’re welcome to send me emails!

Thank you very much!

Sue

## Homework Statement

If I have a collection of data points (vectors), and x and y are two vectors among them. I want to project the data to a direction that the Euclidean distance between x and y is Maximally preserved. Then this direction should be the row space of (x-y)’, denoted as row( (x-y)’ ), right? (suppose column vectors.)
Now, suppose I have n pairs of such data points grouped as a matrix A, say

A = [
(x1 – y1)’
(x2 – y2)’

(xn –yn)’
]

For my problem, directly calculate the row space projection matrix P1 = A’*inv(AA’)*A is difficult, so I want to do some approximation.

## The Attempt at a Solution

Suppose all the vectors have been normalized, that is norm(x1) = norm(x2) = … = norm(xn) = norm(y1) = .. = norm(yn). I define matrix B as

B = [
(x1 + y1)’
(x2 + y2)’

(xn + yn)’
]

And plan to find the null space of B, denoted as null(B). It is easy for me to calculate the null space projection matrix P2 = I – B’*inv(BB’)*B. Note that, (xi – yi)’*(xi+yi) = 0 for normalized vectors.

I feel that null(B) = row(A) . Is it true? Or what’s the relationship between null(B) and row(A)? Can I make the conclusion that, if I project data to null(B), those pair of points (xi, yi) will also be maximally separated?

I’m waiting online. Any input will be appreciated! You’re welcome to send me emails!

Thank you very much!

Sue

## 1. What is a null space projection?

A null space projection is a mathematical method used to find the closest point in a given vector space to a specified point. It is often used in data analysis and machine learning to reduce the dimensionality of data and remove unwanted features.

## 2. How is a null space projection calculated?

The null space projection is calculated by finding the orthogonal projection of the specified point onto the null space of the vector space. This results in a new point that is the closest possible point in the vector space to the specified point.

## 3. What is the purpose of using a null space projection?

The purpose of a null space projection is to simplify and reduce the dimensionality of data. By projecting data onto the null space, unnecessary features can be removed, making the data easier to analyze and interpret.

## 4. How does a null space projection differ from a regular projection?

A regular projection involves finding the closest point in a vector space to a specified point, while a null space projection involves finding the closest point in the vector space that is orthogonal to the specified point. This means that a null space projection takes into account the underlying structure of the data, rather than just the distance between points.

## 5. When is a null space projection commonly used?

A null space projection is commonly used in data analysis and machine learning, particularly in cases where the data has a high dimensionality and needs to be simplified. It is also used in signal processing and image compression to reduce noise and remove redundant features.

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