1. For the following quadratic forms (i) write down the symmetric matrix A (so the form equal to x" Ax), (ii) find matrices P, orthonormal, and D so that D diagonal, (iii) write the quadratic form in terms of y1, decide whether the form is positive/negative definite/semidefinite or indefinite. (a) 2x? – 201x2 + 2x3 (b) –x? + 4x1x2 – 4x2 (c) 3x? + 4.x3 + 5x3 + 4x1x2 – 4x2X3 (it may help to show that A =1 is a root of the characteristic polynomial PT AP is Py and (iv) . ., Yn where x = A).

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