# Why Rank is the Trace of a Projection

• arthurhenry
In summary, a projection operator has eigenvalues of 0 and 1, with the rank being equal to the number of eigenvalues that are 1. The trace of a projection is equal to the sum of all its eigenvalues, which also equals its rank. This information can be found in the book "Algebras of Linear Transformations" by Douglas Farenick."
arthurhenry
Why is the Trace of a projection is its Rank.
Thank you

Hi arthurhenry!

A projection operator P satisfies P2=P. So it's only eigenvalues are 0 and 1. It is easy to see that the rank of P is the number of eigenvalues that are 1. Thus the sum of the eigenvalues is in this case the rank...

Now, take the Jordan normal form of P, then the diagonal contains all the eigenvalues. In particular, the trace is the sum of all the eigenvalues. And thus equals the rank of P.

Dear Micromass,

I thank you for your help --on two occasions now, as you answered another post of mine. Some of these questions come as I verify a comment or at times directly a trying to do an exercise. I am reading a book "Algebras of Linear Transformations" by Douglas Farenick, to teach myself some of that material. I do realize some of my questions are rather rudimentary, I apologize.

Don't apologize! It's only by asking such a questions that you'll learn the material. Everybody has to go through it

for your question. I am happy to provide an explanation for why rank is the trace of a projection and why the trace of a projection is its rank.

First, let's define what a projection is. A projection is a linear transformation that maps a vector space onto itself, such that the image of the transformation is equal to its range. In other words, a projection takes a vector and projects it onto a subspace of the original vector space.

Now, the rank of a projection is the dimension of its range. This means that it represents the number of linearly independent vectors in the subspace that the projection maps to. In other words, it tells us how many dimensions are needed to span the subspace.

On the other hand, the trace of a matrix is the sum of its diagonal elements. In the case of a projection matrix, the diagonal elements represent the eigenvalues of the matrix. And since a projection matrix has only two possible eigenvalues - 0 and 1 - the trace of a projection matrix is simply the number of 1's on the diagonal.

Therefore, we can see that the rank of a projection is equal to the trace of its matrix representation. This is because the number of 1's on the diagonal is equivalent to the number of linearly independent vectors in the subspace that the projection maps to. In other words, the trace of a projection matrix represents the dimension of its range, which is the same as the rank of the projection.

In conclusion, the rank of a projection is the trace of its matrix representation because it represents the number of linearly independent vectors in the subspace that the projection maps to. And the trace of a projection matrix is its rank because it represents the dimension of its range, which is the same as the number of linearly independent vectors in the subspace. I hope this explanation helps to clarify the relationship between rank and the trace of a projection.

## 1. What is the definition of a projection matrix?

A projection matrix is a square matrix that results in the same matrix when multiplied by itself. It represents a linear transformation that projects vectors onto a lower dimensional subspace.

## 2. How is rank defined for a projection matrix?

The rank of a projection matrix is defined as the number of linearly independent rows or columns in the matrix. This is equal to the number of non-zero eigenvalues of the matrix.

## 3. Why is the rank of a projection matrix equal to its trace?

The trace of a matrix is defined as the sum of its diagonal elements. For a projection matrix, the diagonal elements are all either 0 or 1. Thus, the trace of a projection matrix is equal to the number of 1's on its diagonal, which is also the rank of the matrix.

## 4. How does the rank of a projection matrix relate to its null space?

The rank of a projection matrix is equal to the number of dimensions in its image space, which is the complement of its null space. This means that the dimension of the null space is equal to the difference between the dimension of the input space and the rank of the projection matrix.

## 5. Can the rank of a projection matrix be greater than its dimension?

No, the rank of a projection matrix can never be greater than its dimension. This is because the rank is equal to the number of linearly independent rows or columns, and a square matrix cannot have more linearly independent rows or columns than its dimension.

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