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    Big-O notation

    Hi! For the following functions, what are their big-O notation? 1. n^(n-1) 2. (n-1)^n Should their big-O notations be the same as the original functions? i.e. 1. O(n^(n-1)) = n^(n-1)? 2. O((n-1)^n) = (n-1)^n? Please help! Many thanks!
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    Nullity = 1

    For a matrix A, if its nullity is equal to 1, what is the implication of that? What spans its null space? Thanks a lot!
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    Tricky matrix inverse

    Suppose A is a invertible n-by-n matrix. Let B be the inverse of A, i.e. B = A^(-1).It is trivial that A = B^(-1). If we construct a matrix C whose entry is the square of corresponding entry of B, i.e. C_ij = (B_ij)^2, then we compute the inverse of C. We can compute the inverse of C...
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    Complex semidefinite programming

    For semidefinite programming (SDP), we have the primal and dual forms as: primal min <C,X> s.t. <A_i,X> = b_i, i=1,...,m X>=0 dual max <b,y> s.t. y_1*A_1 + ... + y_m*A_m <=C where the data A_i and C are assumed to be real symmetric matrices in many textbooks and online...
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    Eigenvalues of sum of a Hermitian matrix and a diagonal matrix

    Consider two matrices: 1) A is a n-by-n Hermitian matrix with real eigenvalues a_1, a_2, ..., a_n; 2) B is a n-by-n diagonal matrix with real eigenvalues b_1, b_2, ..., b_n. If we form a new matrix C = A + B, can we say anything about the eigenvalues of C (c_1, ..., c_n) from the...
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    Norm question

    Consider a and u are vector of n entries, why the supremum of a dot u subject to the 2-norm of u is less than or equal to r equals r times 2-norm of a, i.e. sup{a.u | ||u||_2 <=r} = r ||a||_2? How can I work out that? Thank you!
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    Express a psd matrix as a sum

    Is it possible to express a n-by-n positive semi-definite matrix (A) in terms of a sum of n terms of something, i.e. A = B1+B2+...+Bn, where each Bi has similar structure? Thanks!
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    Positive semidefinite cone

    Hi! If we have a 2x2 matrix [x y;y z] belonging to a positive semidefinite cone. Why is it equivalent to say x>=0, z>=0, and xz>=y^2? Thanks!