Question

Given the function f(x) = (1/2)x+PT Px+qTx+r, where x € R", PER XH is full rank, then (a) What is Vx.f(x)? (b) Does this function have a minimum, if so: What is the minimum of this function?

Answer 1

For part (a) and (b) see below:

The matrix p is given to be a full rank nx matrix fint lire suppose that pt p _ A. Then the given... expressim from f(). ven by....... fG) = 1 x P p x + x + r. = £ x A x + gtx+r To differentiate, We Calculate the differentiate difference between f (xf h) 4 f(2), where h ER". fauth) z_*() = { ffw"A Qc+W) +g? (3+6) +p |--46 xTAX +g+x+r)

= 1. (TAXE WARENTAL + HAW) +ay+x+nTht" - (1. mTaxt qTx+ r) = {(NTARA WAX +ht An) + mth tops, ... TAn=hlAxia A = ptp is Symmetrie) = ht Axt f g h tt .htah. = ht AxthT N t t . hah. = UT(AntN) + 1 ht Ah. .. Now since the first Summand is linear in h with a tactor (Anton) 4 the Second Simman and is Ghedintie in h et goes to Thursday 0 Fuso faster. Han the first for small de .. so the poste derivative is.AxW.. is f()= płpa tov. Now the matrix pTp is symmetric with hou-hegative eigenvalues, lince p is of full tank, then loch eigen- value of pipis positive. Let , de, .. du are the Eigen values 4 m is the change of basis matrix 4 D = ding. (di) - n. m (PTP/M = D. Then if we use the linear transformatiu X= my in the expression of fal. this leads to f(x) = EPDY &gTM.Ytp. Since the quadratic form YTDY is positive definite , it has the minimum at Yzo (he. Xo) 4 Thas the expresim to has the minimum value for Y o and the minimum value
For better understanding of the problem, we write the expression for a particular matrix P=$\begin{bmatrix} 1 &0 \\ 0 &-2 \end{bmatrix}$ and the calculated the the function f(x) and its derivatives so that one can verify the analytic derivation with the particular one.

See below:

for understanding ( ) 6 )6-2) 6) + (6 (3) . Let promote Them f(A) is given. = f (ny) / 0 () Godtwy tr. o by Get 483 go xx.wp.p+r. ............................ TH pyt ) = 6 -37162,967) ().