Transpose of Matrix as Linear Map

In summary, the relations between a matrix H and its transpose H^T can be understood as linear maps, with H mapping from F^m to F^n and H^T mapping from F^n to F^m. It is possible to make deeper connections between the two, such as the relationship between the kernel and image of H^T and the kernel and image of H. Additionally, HH^T and H^TH can provide insights into the properties of these linear maps. Overall, the row rank being equal to the column rank is a key factor in understanding these relations, as well as the fact that the transpose represents the pullback map on linear functions.
  • #1
chingkui
181
2
What are the relations between a matrix H and its transpose H^T? I am not asking about the relations between the coefficients, I am asking the relations as linear maps (H: F^m->F^n; H^T: F^n->F^m). I am not sure exactly how I should pose the question actually, but I am thinking there is some deeper relations than between their coefficients, like for example, can we say something about the kernel and image of H^T if we know something about the kernel and image of H? What can we say about HH^T: F^n->F^n and H^TH: F^m->F^m?
(F is an arbitrary field)
 
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  • #2
row rank=column rank, that pretty much tells you everything, and the fact that the images and kernels are related to the rows and cols of H's implies soemthing obvious since the row/cols of the transpose are the cols/rows of the original. (i haven't said which way round since it depends on what side you make your matrices act)
 
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  • #3
the transpose represents the pullback map on linear functions.
 

What is the definition of transpose of a matrix as a linear map?

The transpose of a matrix as a linear map is a mathematical operation that involves flipping the rows and columns of a given matrix in order to create a new matrix. This new matrix is considered to be the transpose of the original matrix and represents the same linear transformation, but in a different coordinate system.

How is the transpose of a matrix represented?

The transpose of a matrix is typically represented by adding a superscript "T" to the original matrix. For example, if the original matrix is denoted as A, its transpose would be denoted as AT.

What is the purpose of finding the transpose of a matrix?

Finding the transpose of a matrix can be useful in solving various mathematical problems, such as finding the inverse of a matrix, solving systems of linear equations, and performing vector transformations. It also allows for easier computation and simplification of certain calculations.

What are the properties of the transpose of a matrix?

The transpose of a matrix has several properties, including:

  • The transpose of a transpose is the original matrix.
  • The transpose of a scalar multiple of a matrix is equal to the scalar multiple of the transpose of the original matrix.
  • The transpose of a sum of matrices is equal to the sum of their transposes.
  • The transpose of a product of matrices is equal to the product of their transposes in reverse order.

How does the transpose of a matrix relate to its eigenvalues and eigenvectors?

The transpose of a matrix has the same eigenvalues as the original matrix, but its eigenvectors may be different. However, if the original matrix is symmetric, then the transpose will have the same eigenvectors as the original matrix. This property is known as the spectral theorem.

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