Hermitian matrixPositive semi-definite matrixEigenvaluesHadamard productsThe Schur theorem provides the global bounds for spectrum of the Hadamard product of two positive semi-definite matrices. In this paper, we obtain lower and upper estimations for each eigenvalue of th...
摘要: The Schur theorem provides the global bounds for spectrum of the Hadamard product of two positive semi-definite matrices. In this paper, we obtain lower and upper estimations for each eigenvalue of th关键词: Hermitian matrix Positive semi-definite matrix Eigenvalues Hadamard products ...
the quantity obtained by multiplying the corresponding coordinates of each of two vectors and adding the products, equal to the product of the magnitudes of the vectors and the cosine of the angle between them. Also calleddot product,scalar product. ...
the quantity obtained by multiplying the corresponding coordinates of each of two vectors and adding the products, equal to the product of the magnitudes of the vectors and the cosine of the angle between them. Also calleddot product,scalar product. ...
This inequality includes as a special case the Nudel'man-varcman inequality for the eigenvalues of a product of unitary matrices analogous to Lidskii's inequality for the eigenvalues of a sum of Hermitian matrices.doi:10.1080/03081087408817038...
In many cases, the Hermitian self-orthogonality of the input codes and the assumption that the underlying matrix is unitary can be relaxed. Some special matrices used in the constructions and illustrative examples of good Hermitian self-orthogonal codes have been provided as well. 展开 ...
MosheGoldberg,andRaphaelLoewy 1Introduction LetM n bethesetofn×ncomplexmatrices,andH n bethesetofHermitianmatricesinM n . Inquantumphysics,quantumstatesofasystemwithnphysicalstatesarerepresentedasdensity matricesAinH n ,i.e.,Aispositivesemi-definitewithtraceone;see[7].LetC∈H m andD∈H n be...
The magnitude of a vector x ∈ ℝn is the square root of the inner product of x with itself. ∎ If u and v are vectors in ℝn, then |(u,v)| ≤ ‖u‖ ‖v‖. ∎ An induced inner product on two matrices of the same order is obtained by multiplying corresponding elements ...
[1] For complex matrices, Hermitian means conjugate symmetric, which in the real case reduces to simply symmetric. The theorem of Pólya and Szegö is actually valid for Hermitian matrices, but I simplified the statement for the case of real-valued matrices....
vD_xAD_xv^T=vD_xA(vD_x)^T\geq 0 , 因为 vD_x\in C^n 所以对于A而言找到了特征根,即A psd所以大于0,所以当两个psd matrices share同一个eigen-basis的时候满足相乘依然是psd Case 3 B is of rank r 根据lemma 3.3有 B可以被分解为r个rank one matrices, A\circ B=A\circ(B_1+B_2+\dots...