Problemas de Matrices
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Homework 1 1 Problem 1 Prove range(A) = range(AA 0 ). 2 Problem 2 Let A ∈ R n×n be symmetric. An eigenvector of A is a nonzero vector x ∈ R n such that Ax = λx where λ ∈ C is the corresponding eigenvalue. (a) Prove that all eigenvalues of A are real. (b) Prove that if x and y are eigenvectors corresponding to distinct eigenvalues, then x and y are orthogonal. 3 Problem 3 Let A = A 11 A 12 ··· A 1r A 21 A 22 ··· A 2r . . . . . . . . . . . . A r1 A r2 ··· A rr represent a nonsingular partitioned matrix, where (i,j)th block A ij is of dimensions n i ×n j (i, j = 1, 2,..., r). Partition A -1 as F 11 F 12 ··· F 1r F 21 F 22 ··· F 2r . . . . . . . . . . . . F r1 F r2 ··· F rr where F ij is of the same dimensions as A ij (i, j = 1, 2,..., r). If A is upper block- triangular, then A -1 is upper block-triangular; that is, if A ij = 0 for j < i = 1,..., r, then F ij = 0 for j < i = 1,..., r. Similarly, if A is lower block-triangular, then A -1 is lower block-triangular. Further, if A is (lower or upper) block-triangular, then F ii = A ii -1 ; that is the ith diagonal block of A -1 equals the inverse of the ith diagonal block of A (i = 1,..., r). 1
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ejercicios resueltos de matrices
Transcript of Problemas de Matrices
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Homework 1
1 Problem 1
Prove range(A) = range(AA).
2 Problem 2
Let A Rnn be symmetric. An eigenvector of A is a nonzero vector x Rnsuch that Ax = x where C is the corresponding eigenvalue.
(a) Prove that all eigenvalues of A are real.
(b) Prove that if x and y are eigenvectors corresponding to distinct eigenvalues,then x and y are orthogonal.
3 Problem 3
Let A =
A11 A12 A1rA21 A22 A2r...
.... . .
...
Ar1 Ar2 Arr
represent a nonsingular partitioned matrix,where (i,j)th block Ai j is of dimensions nin j (i, j = 1, 2, . . . , r). Partition A1 as
F11 F12 F1rF21 F22 F2r...
.... . .
...
Fr1 Fr2 Frr
where Fi j is of the same dimensions as Ai j (i, j = 1, 2, . . . , r). If A is upper block-triangular, then A1 is upper block-triangular; that is, if Ai j = 0 for j < i = 1, . . . , r,then Fi j = 0 for j < i = 1, . . . , r. Similarly, if A is lower block-triangular, thenA1 is lower block-triangular. Further, if A is (lower or upper) block-triangular,then Fii = Aii1; that is the ith diagonal block of A1 equals the inverse of the ithdiagonal block of A (i = 1, . . . , r).
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4 Problem 4
Let A11 be an mm matrix, A12 an mn matrix, A21 an nm matrix, and A22 annn matrix. Suppose that the partitioned matrix A =
(A11 A12A21 A22
)is nonsingular,
define B = A1 and partition B as B =(B11 B12B21 B22
), where the dimensions of B11,
B12, B21 and B22 are the same as those of A11, A12, A21 and A22, respectively. IfA11 is nonsingular, then B22 is nonsingular, and
A111 = B11 B12B221B21, A111A12 = B12B221,A21A111 = B221B21, and A22 A21A111A12 = B221.
5 Problem 5
Show that |||A|||22 |||A|||1 |||A||| for all A Rnn.
6 Problem 6
For any A Rmn, show
(a) |||A|||2 ||A||F n||A||2,
(b) 1n|||A||| |||A|||2 m|||A|||,
(c) 1m|||A|||1 |||A|||2 n|||A|||1.
7 Problem 7
If || || is a consistent matrix norm and if F Rnn and ||F|| < 1, then In F isnonsingular; moreover, (In F)1 = k=0 Fk with ||(In F)1|| 11||F|| .8 Problem 8
Let || || be a consistent matrix norm for mm matrices, A and B be mm, then
(a) ||AB Im|| < 1 A and B are nonsingular.
(b) A is nonsingular, B is singular ||A B|| 1||A1 || .
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