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assig11_2011

# assig11_2011 - occupations were divided into three classes...

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Math 152, Spring 2011 Assignment #11 Notes: Each question is worth 5 marks. Due in class: Monday, March 28 for MWF sections; Tuesday, March 29 for TTh sections. Solutions will be posted Tuesday, March 29 in the afternoon. No late assignments will be accepted. 1. Find all eigenvalues and eigenvectors of the matrix A = bracketleftbigg - 4 - 3 6 5 bracketrightbigg 2. Let A be the matrix of the problem above. Denote by v 1 and v 2 the eigenvectors you found for A . (a) Let ˆ e 1 and ˆ e 2 be the standard basis vectors for R 2 . Express ˆ e 1 and ˆ e 2 as linear combinations ˆ e 1 = s 1 v 1 + s 2 v 2 ˆ e 2 = t 1 v 1 + t 2 v 2 of v 1 and v 2 . (b) Recall that if v is an eigenvector of A with eigenvalue λ and if n is any integer, then A n v = λ n v . Use this in conjunction with part (a) to find A 10 ˆ e 1 and A 10 ˆ e 2 . (c) Find A 10 . 3. Find all eigenvalues and eigenvectors of the matrix B = bracketleftbigg 1 - 1 1 1 bracketrightbigg 1

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4. Find a matrix for which bracketleftbig 2 , - 1] T is an eigenvector of eigenvalue 2 and bracketleftbig
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Unformatted text preview: occupations were divided into three classes U: professional, managerial, executive M: inspectional, supervisory, other non–manual, skilled manual L: semi–skilled and unskilled manual The study collected data relating the occupational classes of sons to that of their fathers (only males were considered.) It found, for exam-ple, that 5.4% of sons of M–class fathers ended up in U–class occupa-tions. The full transition matrix was T = U M L U . 448 . 054 . 011 M . 484 . 699 . 503 L . 068 . 247 . 486 Find a vector x , having positive components that add up to 1, that obeys T x = x . Such a vector is called an equilibrium or steady state. 2...
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assig11_2011 - occupations were divided into three classes...

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