Advanced Linear Algebra Problems

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กระทู้นี้สำหรับคนชอบ Algebra แบบ advanced ครับ
ชุดนี้เอาโจทย์ linear algebra ที่เป็นข้อสอบ qualify มาลงครับ
ขอเป็นภาษาอังกฤษนะครับเพราะศัพท์เทคนิคเยอะเหลือเกิน

1. (UMCP August 2004) Let A be an invertible square matrix over C. Suppose Aⁿ is diagonalizable for some nณ1. Show that A is diagonalizable.

2. (UMCP January 2004) Let A and B be nonzero nxn matrices over C. Suppose that AB=BA. Show that if the characteristic polynomial of A is separable(no multiple roots) then the minimal polynomial of B is separable.

3. (UMCP August 2003) Let V be a finite dimensional vector space over a field F. Let T:V-->V be a linear transformation.
(a) Suppose that every nonzero vฮV is an eigenvector of T. Show that T is a scalar multiple of the identity.
(b) A cyclic vector for T is a vector vฮV such that {v,Tv,T²v,...} spans V. Suppose that every nonzero vector vฮV is a cyclic vector for T. Show that the characteristic polynomial of T must be irreducible over F.
(c) Suppose that V is 2-dimensional over F and that the characteristic polynomial of T is irreducible over F. Show that every nonzero vฮV is a cyclic vector for T.

4. (UMCP January 2003) Let M be an mxn matrix over a field F. Show that there exist an invertible mxm matrix A and invertible nxn matrix B such that AMB = (c_ij), with c_ii = 0 or 1 for all i and c_ij = 0 whenever iนj.

5. (UMCP August 2002) Let A be an mxn matrix with real entries, where mณn. Assume A has rank n. Let xทy = x^Ty be the standard dot product on R^m.
(a) Show that A^TA is invertible. (Hint : Rewrite AxทAx)
(b) Let P = A(A^TA)^-1A^T. Show that (x - Px)ทPx = 0 for all xฮR^m