Explicit Block Diagonal Decomposition of Block Matrices Corresponding to Symmetr
Explicit Block Diagonal Decomposition of Block Matrices Corresponding to Symmetr
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Then all A^j have the same modal matrix V, i. E. , A. J = VN-jV""", i, j = 1, . .. , n (2. 1) where Nij = r Vij, J';^^ ^ V = [y, X, . =^, v-^ = [%, ]^, ,=i (2. 2; Thus ;. A, = [VN^. V-Mjj^i = (I^xV)N, (I^xV-b . Where li^ is a full block matrix, but its blocks are diagonal and "x" is the Kronecker (tensor) multiplication. Therefore one can write A» = [(I^xV)P^] [pIn^P^] [P, (l^xv"b] = X^N^x;^ (2. ^) -7- where N» is block diagonal with blocks of order n N* = f N, J-^, = [ [v, jj?^. ^, J-^^ (2. . ...) and X# is a full block matrix with rectangular mxn blocks THEOREM 2. 2 : If all blocks A^^^ of A» (1. 5) commute and therefore they have the following explicit spectral decomposition ^T = "^ot""^ • 0'^=^ "^ (2. 7) where "oT= ^^Jo. JJ. , . U= [u, ^]l^. , . 0-1 =[uij]^. J. I, (2. 8) then matrix k^ (1. ^) has the following explicit block diagonal decomposition: A^ = U»M^U;1, (2. 9) with ^* = i ^J J^l = ^ ^JctIo. T^I Jj = 1 • (2. 10) PROOF: According to Theorem 2. 1 Therefore A* = P»A»pi = (P^Y^)M»(Y;'pi) = U, M»U;' with (2.
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To quickly assess the difficulty of the text, read a short excerpt:
Then all A^j have the same modal matrix V, i. E. , A. J = VN-jV""", i, j = 1, . .. , n (2. 1) where Nij = r Vij, J';^^ ^ V = [y, X, . =^, v-^ = [%, ]^, ,=i (2. 2; Thus ;. A, = [VN^. V-Mjj^i = (I^xV)N, (I^xV-b . Where li^ is a full block matrix, but its blocks are diagonal and "x" is the Kronecker (tensor) multiplication. Therefore one can write A» = [(I^xV)P^] [pIn^P^] [P, (l^xv"b] = X^N^x;^ (2. ^) -7- where N» is block diagonal with blocks of order n N* = f N, J-^, = [ [v, jj?^. ^, J-^^ (2. . ...) and X# is a full block matrix with rectangular mxn blocks THEOREM 2. 2 : If all blocks A^^^ of A» (1. 5) commute and therefore they have the following explicit spectral decomposition ^T = "^ot""^ • 0'^=^ "^ (2. 7) where "oT= ^^Jo. JJ. , . U= [u, ^]l^. , . 0-1 =[uij]^. J. I, (2. 8) then matrix k^ (1. ^) has the following explicit block diagonal decomposition: A^ = U»M^U;1, (2. 9) with ^* = i ^J J^l = ^ ^JctIo. T^I Jj = 1 • (2. 10) PROOF: According to Theorem 2. 1 Therefore A* = P»A»pi = (P^Y^)M»(Y;'pi) = U, M»U;' with (2.
What to read after Explicit Block Diagonal Decomposition of Block Matrices Corresponding to Symmetr?
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