Modern Characterization of Electromagnetic Systems and its Associated Metrology. Magdalena Salazar-Palma

side of (1.28) over to the left side of the equation and equate it to zero as such

(1.31)

If the concatenated matrix [X y] has a rank of n + 1, the n + 1 columns of the matrix are linearly independent and the n + 1, m‐dimensional columns of the matrix span the same n‐dimensional space as X. In order to have a unique solution for the coefficients, a, the matrix [X + ; y + ] must have n linearly independent columns. However, this matrix has n + 1 columns in total and therefore is rank is deficient by 1. We then must find the smallest matrix [] that changes matrix [X y] with a rank of n + 1, to a matrix {[X y] + []} with a rank n. According to the Eckart‐Young‐Mirsky theorem we can achieve this by defining {[X y] + []} as the best rank‐n approximation to [X y] and by eliminating the last singular value of [X y] which contains the least amount of system information and provides a unique solution. The Eckart–Young–Mirsky theorem (https://en.wikipedia.org/wiki/Low‐rank_approximation) states a low‐rank approximation is a minimization problem, in which the cost function measures the fit between a given matrix (the data) and an approximating matrix (the optimization variable), subject to a constraint that the approximating matrix has reduced rank. To illustrate how this is accomplished, we take the SVD of [X y] as follows

(1.32)

      where U_x has n columns, u_y is a column vector, ∑_x contains the n largest singular values diagonally, σ_y is the smallest singular value, V_xx is a n × n matrix, and v_yy is scalar. Let us multiple both sides by matrix V.

      (1.33)

Next, we will equate just the last columns of the matrix multiplication occurring in (1.32).

(1.34)

      From the Eckart‐Young theorem, we know that {[X y] + []} is the closest rank‐n approximation to [X y]. Matrix {[X y] + []} has the same singular vectors contained in ∑_x above with σ_y equal to zero. We can then write the SVD of {[X y] + []} as such

(1.35)

      To obtain [] we must solve the following

(1.36)

(1.36) can be solved by first using (1.32) and (1.35) which results in

(1.37)

Then, from (1.34) we can rewrite (1.37) as

      (1.38)

      Finally, {[X y] + []} can be defined as

(1.39)

After multiplying each term in (1.39) by we get the following

      (1.40)

      The right‐hand side cancels and we are left with

(1.41)

From (1.31) and (1.41) we can solve for the model coefficient a as

      (1.42)

      The vector v_xy is the first n elements of the n +1‐th columns of the right singular matrix V, of [X y] and v_yy is the n + 1‐th element of the n + 1 columns of V. The best approximation of the model is then given by

      (1.43)

      This completes the total least squares solution.

1.4 Conclusion

      This first chapter provides the mathematical fundamentals that will be utilized later. The principles presented are the basis of low‐rank modelling, singular value decomposition and the method of total least squares.

References

      1 1 L. I. Scharf and D. Tufts, “Rank Reduction for Modeling Stationary Signals,” IEEE Transactions on Acoustics,