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10 Nov 2019
1 4 -8 8. (11 points) Consider the symmetric matrix B-4 7 4 which can be orthogonally -8 4 1 -9 0 0 diagonalized as: B-13 2/3 2/3 0 9 0-1/3 2/3 2/3 2/3 2/3-1/31 0 0 9 2/3 2/3-1/3 a. What is the minimum of the quadratic form Q(x) of B. over all unit length vectors x? b. What is the maximum of the quadratic form Q(x) of B, over all unit length vectors x? c. For what unit vectors x will Q(x) take its maximum value? d. For what unit vectors x will Q(x) take its minimum value? e. Find a unit vector x at which (x)0. Describe the set of unit vectors x at which Q(x)-0.
1 4 -8 8. (11 points) Consider the symmetric matrix B-4 7 4 which can be orthogonally -8 4 1 -9 0 0 diagonalized as: B-13 2/3 2/3 0 9 0-1/3 2/3 2/3 2/3 2/3-1/31 0 0 9 2/3 2/3-1/3 a. What is the minimum of the quadratic form Q(x) of B. over all unit length vectors x? b. What is the maximum of the quadratic form Q(x) of B, over all unit length vectors x? c. For what unit vectors x will Q(x) take its maximum value? d. For what unit vectors x will Q(x) take its minimum value? e. Find a unit vector x at which (x)0. Describe the set of unit vectors x at which Q(x)-0.
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