The Transpose of a Matrix

In summary, for the given equation B=A^{T}A, the elements b_{ij} can be found by taking the dot product of row i and column j of the matrix A. This can also be represented in matrix notation as b_{ij}=a^{T}_{i}a_{j}.
  • #1
jtruth914
21
0
Let [itex]B=A^{T}A[/itex]. Show that [itex]b_{ij}[/itex]=[itex]a^{T}_{i}[/itex][itex]a_{j}[/itex].

I have no idea how to approach this problem.
 
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  • #2
jtruth914 said:
Let [itex]B=A^{T}A[/itex]. Show that [itex]b_{ij}[/itex]=[itex]a^{T}_{i}[/itex][itex]a_{j}[/itex].

I have no idea how to approach this problem.
I don't understand your notation. Could you please clarify what you've written?
 
  • #3
Let [itex]B=A^{T}A[/itex]. Show that [itex]b_{ij}[/itex]=[itex]a^{T}_{i}[/itex][itex]a_{j}[/itex].

When you transpose [itex]A[/itex]You are flipping the rows and columns.
When you multiply [itex]A^{T}A[/itex] you would generate each element [itex]b_{ij}[/itex] will be the dot product of row i of the first matrix and column j of the second matrix. Since the first matrix is the transposition of the second, row i of that transposition will be column i of the original. So each element will be the dot product of two column vectors:

[itex]b_{ij}[/itex]=[itex]a_{i}[/itex]·[itex]a_{j}[/itex]

Using matrix notation, this is:

[itex]b_{ij}[/itex]=[itex]a^{T}_{i}[/itex][itex]a_{j}[/itex]
 

What is the definition of the transpose of a matrix?

The transpose of a matrix is a new matrix that is created by exchanging the rows and columns of the original matrix. This means that the first row of the original matrix becomes the first column of the transposed matrix, the second row becomes the second column, and so on.

How is the transpose of a matrix written mathematically?

The transpose of a matrix A is written as AT.

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

The transpose of a matrix is useful in many areas of mathematics and science, including linear algebra, statistics, and computer graphics. It can help simplify calculations and solve equations more easily.

What is the difference between a square matrix and its transpose?

A square matrix has an equal number of rows and columns, while its transpose has the rows and columns switched. This means that a square matrix and its transpose have the same dimensions, but their elements are arranged differently.

How is the transpose of a matrix calculated?

To calculate the transpose of a matrix, simply write the elements of the original matrix as the elements of the transposed matrix, but switch the rows and columns. For example, if the original matrix has a row of [1 2 3], the transposed matrix will have a column of [1; 2; 3].

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