The matrix that sends b to bˆ is [tex]A(A^{T} A)^{-1} A^{T} b[/tex]
A collection of integers lined up in rows and columns to form a rectangular array is called a matrix. The elements, or entries, of the matrix, are the integers. In addition to several mathematical disciplines, matrices find extensive use in the fields of engineering, physics, economics, and statistics.
Given Ax = b, we know that xˆ solves the least squares problem:
xˆ [tex]= (A^{T} A)^{-1} A^{T} b[/tex]
And that bˆ = Axˆ. Therefore, we get bˆ by multiplying both sides of our previous equation by A:
bˆ = Axˆ [tex]= A(A^{T} A)^{-1} A^{T} b[/tex]
Therefore, the matrix that sends b to bˆ is [tex]A(A^{T} A)^{-1} A^{T} b[/tex].
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