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| m58620 | By definition, A is idempotent if A2 = A, and B is nilpotent if Bm = 0 for some positive integer m. Give examples (different from 0 or I). Also give examples such that A2 = I (the unit matrix). |
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| m58625 | (c) Circle. Fine a similar formula for a circle in the plane through three given points. Find and sketch the circle through (2, 6), (6, 4), (7, 1).
(d) Sphere. Find the analog of the formula in (c) for a sphere through four given points. Find the sphere through (0, 0, 5), (4, 0, 1), (0, 4, 1), (0, 0, – 3) by this formula or byinspection. |
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| m58629 | Calculate AB in Prob. 1 column wise. (See Example 6) |
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| m58648 | Find the Eigen values and Eigen vectors of the so-called Pauli spin matrices and show that SxSy = iSz, SySx = – iSz, Sx2 = Sy2 = Sz2 =I, |
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| m58925 | Find all 2 x 2 matrices A = [ajk], B = [bjk], C = [cjk] and general scalar. |
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| m58943 | Find all vectors v in R3 orthogonal to [2 0 1]T. |
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| m58948 | Find and Eigen basis and diagonalize, (Show thedetails). |
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| m58949 | Find and Eigen basis and diagonalize, (Show thedetails). |
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| m58950 | Find and Eigen basis and diagonalize, (Show thedetails). |
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| m58962 | Find Eigen vectors of A, B, C in Examples 2 and 3. |
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| m59079 | Find the Eigen values and Eigen vectors of the so-called Pauli spin matrices and show that SxSy = iSz, SySx = – iSz, Sx2 = Sy2 = Sz2 =I, |
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| m59172 | Find the value of the determinant of the n x n matrix and with main diagonal entries all 0 and all others 1. Try to find a formula for this. Try to prove it by induction. Interpret A3 and A4 as "incidence matrices" (as in Problem Set 7.1 but without the minuses) of a triangle and a tetrahedron, respectively; similarly for an "n-simplex", having n vertices and n (n – 1)/2 edges (and spanning Rn–1, n = 5, 6 . . .) |
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| m59326 | Give an application of the matrix in Prob. 3 that makes the form of its inverse obvious. |
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| m59431 | If a ray of light is reflected once in each of two mutually perpendicular mirrors, what can you say about the reflected ray? |
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| m59541 | If |p| = 1 and |q| = 2, what can be said about the magnitude and direction of the resultant? Can you think of an application where this matters? |
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| m60147 | Prove the following statements and illustrate them with examples of your own choice. Here, λ1, ∙ ∙ ∙ λn are the (not necessarily distinct) Eigen values of a given n x n matrix A = [ajk].
(a) Trace the sum of the main diagonal entries is called the trace of A. It equals the sum of the Eigen values.
(b) “Spectral shift”. A – k1 has the Eigen values λ1 – k, ∙∙∙ λn – k and the same Eigen vectors as A.
(c) Scalar multiples, powers. kA has the Eigen values k λ1, ∙∙∙ kλn. Am (m = 1, 2, ∙∙∙) has the Eigen values λ1m, ∙∙∙ λnm. The Eigen vectors are those of A. (d) Spectral mapping theorem, the "polynomial matrix"
Has the Eigen values where j = 1, ∙∙∙ nm and the same Eigen vectors as A.
(e) Peron’s theorem, Show that a Leslie matrix L with positive l12, l13, l21, l32, has a positive Eigen value. (This is a special case of the famous Perron-Frobenius theorem in Sec. 20.7, which is difficult to prove in its general form.). |
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| m60150 | Prove the formula in Prob. 15. |
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| m60151 | Prove the formula in Prob. 17. |
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| m60181 | Reduce the quadratic form to principal axes.
11.56x12 + 20.16x1 x2 + 17.44x22 = 100 |
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| m60182 | Reduce the quadratic form to principal axes.
14x12 + 24x1 x2 – 4x22 = 20 |
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