# Projections and diagonal form (Algebra)

• math8
In summary, if V is a vector space with finite dimension n, then two projections T and P have the same diagonal form if and only if their kernels have the same dimension. This can be proven using the formula dimV = dim ker(T) + dim Im(T) and by forming a basis of V from vectors in ker(T) and Im(T). This also shows that the matrices of T and P are similar, thus leading to the same diagonal form.
math8
If V has finite dimension n, show that two projections have the same diagonal form if and only if their kernels have the same dimension. (A projection is defined to be a linear transformation P:V-->V for which P^2=P; V is a vector space).

For the forward direction, I was thinking that if T and P are projections that have the same diagonal form, their matrices of transformation are similar.
I think I need to use the formula
dimV= dim ker(T)+ dim Im(T) Knowing that dim Im(T)= rank of the matrix of transformation of T and dim ker(T)= n-dim Im(T)
So the result follows.

But I am not sure about the backward direction.

Form a basis of V from vectors in ker(T) and im(T). You know the two sets span V. That's where your dimension formula comes from. Think about what the matrix of T looks like in this basis.

Since the dim of the kernels are the same, then the dim of the Images are the same as well. We also know that for a projection V=ker T '+' I am T (where '+' denotes the direct sum)
So if {e1,e2,...,ek} is a basis for ker T, we may complete this list by vectors of the basis for I am T to get a basis for V ({e1,e2,...,ek,f1,...,fm}.

Now the matrix of the transformation T with respect to this new basis is a k+m X k+m matrix. I don't know what else to say.

T(ei)=0, T(fi)=fi. It looks to me like that means you have a matrix with 0's and 1's on the diagonal. How many zeros and how many ones?

there should be k 0's and m 1's.

So the same thing for both the projections T and P, hence they have the same diagonal form?

Ok, so k is dim(ker(T)) and m is dim(V)-dim(ker(T)). If you have two diagonal matrices with the same number of 1's and 0's on the diagonal, are they similar?

math8 said:
So the same thing for both the projections T and P, hence they have the same diagonal form?

Sure. They have the same matrix in different bases. They are similar.

Thanks, it makes much more sense now.

## 1. What is a projection in algebra?

A projection in algebra refers to a linear transformation that maps a vector space onto a subspace of that space. It essentially "projects" a vector onto a lower-dimensional space, preserving its direction but possibly changing its magnitude.

## 2. How is a projection represented in diagonal form?

A projection can be represented in diagonal form by using an orthogonal basis for the subspace onto which the projection is being made. In this case, the projection matrix will have 1's along the diagonal and 0's everywhere else.

## 3. What is the difference between orthogonal and non-orthogonal projections?

The main difference between orthogonal and non-orthogonal projections lies in the properties of the projection matrix. An orthogonal projection matrix is symmetric, idempotent, and has eigenvalues of 0 and 1, while a non-orthogonal projection matrix may not have these properties.

## 4. How is the concept of projections applied in real-world scenarios?

Projections are widely used in fields such as engineering, physics, and computer graphics to represent and manipulate data. In engineering, projections are used to simplify complex systems by projecting them onto lower-dimensional subspaces. In computer graphics, projections are used to map 3D objects onto a 2D screen, creating the illusion of depth.

## 5. What are some common applications of diagonal form in linear algebra?

Diagonal form is commonly used in linear algebra to simplify computations and solve systems of equations. It can also be used to find the eigenvalues and eigenvectors of a matrix, which have various applications in physics, engineering, and other fields. Additionally, diagonal form is used in statistics for principal component analysis, a technique for reducing the dimensionality of data.

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