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Yagoda
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Homework Statement
I want to show that for an n x n matrix A with complex entries, if [itex]\left\|Ax\right\|=\left\|x\right\|[/itex] for any vector x in C^n, then the rows of A are an orthonormal basis of C^n.
Yes. If the row vectors of A are a1..an and x = a1T, you have <a1,a1>2+<a1,a2>2+... You know that <a1,a2>2 etc are non-negative, so...?Yagoda said:Okay, suppose I'm setting x to be the first columns of the adjoint of A. Then when I take <Ax,Ax> the first term is <x,x>*<x,x> and the rest of the terms are the inner product of the first row of A with another term of A.
The norm of a complex matrix is a measure of its magnitude or size. It is calculated by finding the square root of the sum of squares of all the elements in the matrix. In other words, it represents the length of the longest vector that can be formed by multiplying the matrix with a vector of appropriate dimensions.
The norm of a complex matrix takes into account the magnitude of both the real and imaginary components of the matrix, whereas the norm of a real matrix only considers the magnitude of the real components. This means that the norm of a complex matrix can be larger than the norm of a real matrix even if they have the same number of elements.
The norm of a complex matrix is used to measure the error or difference between two matrices. It is also a useful tool for determining the convergence of iterative algorithms and for solving optimization problems.
No, the norm of a complex matrix is always a positive value. This is because it is calculated by taking the square root of the sum of squares, which always results in a positive value.
The rows of a complex matrix are the horizontal lines of elements that make up the matrix. Each row represents a different vector, and together they form the basis for the entire matrix. The number of rows in a matrix is equal to its number of dimensions.