Determine the definiteness of the quadratic form given by Q(x,y,z) = (x + y - z)^2 (mark all that applies)

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The quadratic form given is Q(x,y,z) = (x + y - z)^2.

Option 1: negative definite. A negative definite form would satisfy Q(x) < 0 for all nonzero x. Here, Q is a square, so Q(x,y,z) ≥ 0 for all inputs, and it is impossible for Q to be negative. Therefore, this option is incorrect.

Option 2: negative semi-definite. A negative semi-definite form would satisfy Q(x) ≤ 0 for all x, with equality only possibly at x = 0. Since Q(x,y,z) is a square, it is nonnegative for all x, and there exist nonzero vectors where Q > 0 (e.g., x=1, y=z=0 gives Q=1). Thus, this option is also incorrect.

Option 3: indefinite. An indefinite form takes on both positive and negative values. Our Q is never negative, so it cannot be indefinite. This option is incorrect.

Option 4: positive semi-definite. Here, Q(x,y,z) ≥ 0 for all (x,y,z), and Q can be zero for nonzero vectors precisely when x + y - z = 0. Since there are nonzero inputs with Q = 0, the form is not positive definite but is positive semi-definite. This option is correct.

Option 5: positive definite. Positive definiteness would require Q(x) > 0 for all nonzero x. That fails because any nonzero vector with x + y - z = 0 yields Q = 0. Hence, this option is incorrect.

Dashboard Online Test 3, April 27

Determine the definiteness of the quadratic form given by Q(x,y,z) = (x + y - z)^2 (mark all that applies)

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