Comments on “ Is the Frobenius Matrix Norm Induced ? ”
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In “Is the Frobenius Matrix Norm Induced?”, the authors ask whether the Frobenius and the norms are induced. There, they claimed that the Frobenius norm is not induced and, consequently, conjectured that the norm may not be induced. In this note, it is shown that the Frobenius norm is induced on particular matrix spaces. It is then shown that the norm is in fact induced on a particular matrix-valued space. NOTATION , Field of real and complex numbers, respectively. n n-dimensional real space. n n-dimensional complex space. m n Space of m n matrices with real entries. m n Space of m n matrices with complex entries. A Complex conjugate transpose of A. tr(A) Trace of A. kAkF Frobenius norm of A. kAkF = tr(AA?) = i 2 i where i’s singular values of the matrix A; Aij (i; j)th element of A; jAj largest singular value of the matrix A; I identity matrix; ess sup abbreviation for essential supremum; L2 m n space of Lebesgue square integrablem n matrix-valued functions defined on j!-axis, with the following norm:
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Authors' reply - Comments on "Is the frobenius matrix norm induced?"
In a paper [1] the authors ask whether the Frobenius and the H norms are induced. There they claimed that the Frobenius norm is not induced, and consequently conjectured that the H-norm may not be induced. In this note it is shown that the Frobenius norm is induced on particular matrix spaces. It is then shown that the H-norm is in fact induced on a particular matrix-valued L1 space. NOTATION R...
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