v2007.09.13 - Convex Optimization

512 APPENDIX A. LINEAR ALGEBRATherefore diagonalizable A has rank(A) nonzero eigenvalues and exactlym−rank(A) eigenvalues equal to 0 whose corresponding eigenvectors spanN(A).By similar argument, the left-eigenvectors corresponding to 0 eigenvaluesspan N(A T ).Next we show when A is diagonalizable, the real and imaginary parts ofits eigenvectors (corresponding to nonzero eigenvalues) span R(A) :The right-eigenvectors of a diagonalizable matrix A∈ R m×m are linearlyindependent if and only if the left-eigenvectors are. So, matrix A hasa representation in terms of its right- and left-eigenvectors; from thediagonalization (1334), assuming 0 eigenvalues are ordered last,A =m∑λ i s i wi T =i=1k∑≤ mi=1λ i ≠0λ i s i w T i (1367)From the linearly independent dyads theorem (B.1.1.0.2), the dyads {s i w T i }must be independent because each set of eigenvectors are; hence rankA=k ,the number of nonzero eigenvalues. Complex eigenvectors and eigenvaluesare common for real matrices, and must come in complex conjugate pairs forthe summation to remain real. Assume that conjugate pairs of eigenvaluesappear in sequence. Given any particular conjugate pair from (1367), we getthe partial summationλ i s i w T i + λ ∗ i s ∗ iw ∗Ti = 2Re(λ i s i w T i )= 2 ( Res i Re(λ i w T i ) − Im s i Im(λ i w T i ) ) (1368)where A.18 λ ∗ i = λ i+1 , s ∗ iequivalently writtenA = 2 ∑ iλ ∈ Cλ i ≠0∆∆= s i+1 , and w ∗ iRes 2i Re(λ 2i w T 2i) − Im s 2i Im(λ 2i w T 2i) + ∑ jλ ∈ Rλ j ≠0∆= w i+1 . Then (1367) isλ j s j w T j (1369)The summation (1369) shows: A is a linear combination of real and imaginaryparts of its right-eigenvectors corresponding to nonzero eigenvalues. The kvectors {Re s i ∈ R m , Im s i ∈ R m | λ i ≠0, i∈{1... m}} must therefore spanthe range of diagonalizable matrix A .The argument is similar regarding the span of the left-eigenvectors. A.18 The complex conjugate of w is denoted w ∗ , while its conjugate transpose is denotedby w H = w ∗T .