四 ↓an Figure 2: The matrices Bn and CN when n>N of sample covariance matrices that in many engineering and statistical applications. However, among other things, it is subtly different be of the normalization of the by the number of rows N of YN rather than by the number of colum Hence, we need to come up definition of a sample covar matrix that mirrors the manner in which it is used in practical applications With engineering, particularly signal processing applications in mind, we introduce the n x N matrix In'-Xn and define to be the sample covariance matrit(SCM). By comparing(18)and(19) while recalling the definitions of YN and On, it is clear that the eigenvalues of Bn are related to the eigenvalues of Cn. For BN of the form in(18) the Marcenko- Pastur theorem can be used to obtain the canonical equation for mB(a) given by(13). Recal that by mB(z) we mean the Stieltjes transform associated with the limiting e d f. F(r) of BN as N-00 There is however, an exact relationship between the non-limiting e df's FBN() and the FCn(a)and hence the corresponding Stieltjes transforms mBN(a) and mc, (z) respectively. We exploit this relationship below to derive the canonical equation for Cn from the canonical equation for BN given in(13) Figure 2 schematically depicts Cn and BN when n>N i.e. when c>l. In this case, Cn, as denoted in the figure, will have n-N zero eigenvalues. The other N eigenvalues of Cn will, however, be identically equal to the N eigenvalues of BN. Hence, the e d f. of Cn can be exactly expressed in terms of the e.d. f of Recalling the definition of the Stieltjes transform in(3), this implies that the Stieltjes transform mcn (a) of Cn is related to the Stieltjes transform mBN(z) of BN by the expression mc,(2) +cmB(2) Similarly, Figure 3 schematically depicts Cn and BN when n< N i.e. c<l. In this case, BN, as denoted in the figure, will have N-n zero eigenvalues. The other n eigenvalues of BN will, however, be identically equal to the n eigenvalues of Cn. Hence, as before, the e.d. f. of BN can be exactly expressed in terms of the e.d.f. of C. as N(a) I(o,oo+-FCn(r 2-1)4as+=Fc Once again, recalling the definition of the Stieltjes transform in(3), this implies that the Stieltjes transform mcn(=)of Cn is related to the Stieltjes transform mBN(a) of Bn by the expres� � � � � � � � 1 n n X∗ n n N n N n T 1/2 n n T 1/2 Xn n n N n n n n N n X∗ n T 1/2 T n Xn 1/2 n O 1 N n n N YN × YN × O ∗ ∗ = = Cn BN Figure 2: The matrices Bn and CN when n > N. of sample covariance matrices that appear in many engineering and statistical applications. However, among ∗ other things, it is subtly different because of the normalization of the YN YN by the number of rows N of YN rather than by the number of columns n. Hence, we need to come up with a definition of a sample covariance matrix that mirrors the manner in which it is used in practical applications. With engineering, particularly signal processing applications in mind, we introduce the n × N matrix, 1/2 On = Tn Xn and define 1 ∗ Cn = OnOn (19) N to be the sample covariance matrix (SCM). By comparing (18) and (19) while recalling the definitions of YN and On, it is clear that the eigenvalues of Bn are related to the eigenvalues of Cn. For BN of the form in (18), the Marˇcenko-Pastur theorem can be used to obtain the canonical equation for mB(z) given by (13). Recall, that by mB(z) we mean the Stieltjes transform associated with the limiting e.d.f. F B(x) of BN as N → ∞. There is however, an exact relationship between the non-limiting e.d.f.’s F BN (x) and the F Cn (x) and hence the corresponding Stieltjes transforms mBN (z) and mCn (z) respectively. We exploit this relationship below to derive the canonical equation for Cn from the canonical equation for BN given in (13). Figure 2 schematically depicts Cn and BN when n > N i.e. when c > 1. In this case, Cn, as denoted in the figure, will have n − N zero eigenvalues. The other N eigenvalues of Cn will, however, be identically equal to the N eigenvalues of BN . Hence, the e.d.f. of Cn can be exactly expressed in terms of the e.d.f. of BN as n F Cn (x) = n − N I(0,∞] + F BN (x) (20) N N = (c − 1)I(0,∞] + cF BN (x). (21) Recalling the definition of the Stieltjes transform in (3), this implies that the Stieltjes transform mCn (z) of Cn is related to the Stieltjes transform mBN (z) of BN by the expression mCn (z) = − c − 1 + cmBN (z). (22) z Similarly, Figure 3 schematically depicts Cn and BN when n < N i.e. c < 1. In this case, BN , as denoted in the figure, will have N − n zero eigenvalues. The other n eigenvalues of BN will, however, be identically equal to the n eigenvalues of Cn. Hence, as before, the e.d.f. of BN can be exactly expressed in terms of the e.d.f. of Cn as N F BN (x) = N − n I(0,∞] + F Cn (x) (23) n n 1 1 = c − 1 I(0,∞] + F Cn (x) (24) c Once again, recalling the definition of the Stieltjes transform in (3), this implies that the Stieltjes transform mCn (z) of Cn is related to the Stieltjes transform mBN (z) of BN by the expression 1 1 1 mBN (z) = − c − 1 z + c mCn (z). (25)