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研究所、轉學考(插大)-基礎數學
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112年 - 112 國立臺北大學_碩士班入學考試_統計學系﹕基礎數學#130312
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題組內容
3.
(a)
其他申論題
Problem 5 :Let A, B ∈ (R). Prove that rank A + rank B ≤ n if and only if there exists an invertible matrix X ∈ (R) such that AXB = .
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Problem 6 :Let A and B be elements in (C). Suppose that AB - BA = c⋅(A - B) for some non-zero c ∈ C. Prove that there exists an invertible matrix P ∈ (C) such that AP and BP are upper-triangular matrices with the same diagonal entries.
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一、CALCULUS 1.
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2.
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4. Where μ and , i = 1,2,...,n are known values. Find the value of σ at which L(σ) has its maximum. (Skip any derivative test)
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(a) Please write down the (i,j)-entry of the product BCA in terms of Sigma notations (i.e. Σ).
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(b) Please prove that .
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(a) Write down the characteristic polynomial of A and use it to find the eigenvalues.
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(b) Find the eigenspaces of A.
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(c) Orthogonally diagonalize the matrix A. (You need to find out an orthogonal matrix P and a diagonal matrix D such that PᵀAP = D.)
#552132
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