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matrixalgebra - 448 ST.0114 BE 3 2 ST 4831 PS 1 1 ST...

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Unformatted text preview: 448 ST .0114 BE 3 2 ST 4831 PS 1 1 ST 452.711 95 2 2 ST 13.656 PS 1 ST .165 PS sT .209 PS sT .282 1) ST .1211 P on SE Tv 2 33 44 555 366 RS M1 TO N5 llll. un.1~u.nn.. mu”. .., . (1982), Graybill (1983), or Hadley (1961). . covariance between x1 and x2,COV(x1, ‘ and the regression coefficient ,821. ‘ row or column vector. ‘ indicates one row and c I column vector is r X 1. and a’ and I" are row vect prime superscript distingui unneborg and Abbott (1983), Searle , SCALARS, VECTORS, AND MATRICES en scalars, vectors, and matrices. A single eferred to as a scalar. For instance, the x2), is a scalar, as is the number 5 When two or more scalars are written in a row or a column, they form a ow vector is 1 X c, where the 1 is the number of columns (elements); the order of a Three examples of vectors are The order of a r ors of orders 1 X 4 and 1 X 2, respectively. The shes row vectors (e.g., a’) from column vectors 449 50 MA'i‘Kix ALGEBRA RLVIEW 4 (e g E). The prime stands for the transpose operator, which I Will dehne later. . , ( ‘ ‘ A matrix is a group of elements arranged into rows and columns. Man-ICCS (”50 die [Cplcsenled by boldface Syllll)()l5. ”16 ()lthl of d nldtllx 15 I ‘ ' ' V ml)Cr 0‘ L [ed l) I C WllClC I IS the numl)t:t 0t IOWS And ( l5 lllC nu lndl a X , . _ , ' I \ C lumnS. [>116 OldCl r X C 3 ISO '5 [he lmcns‘lOll of {he IIMIHX. A5 delllplcb, .0 I ‘ d . consider 0 712 S : [3'11 512], F : y” (l . .1 ‘ .y . 7 . 'x. The dimension of S is 2 X 2, whereas I‘ fis a 3 ‘X .. graft; vector is a 1 X (‘ ‘ ' ' ‘ ' ‘ ecial cases 0 matrices. . Vectors and scalars are sp . . H . t» A mamx- 1 trix a column vector is an r X 1 matrix, and a scalar is a 1 X11 d .f n a i 4 ' " an I TW) matrices say S and T are equal if they are of the sameforrtfer 1| 45 i , ‘ r ' l ‘ ' t t 0 or a 1 every element 51 of S equals the corresponding elemen d ,+* W “m. / V r x A . A 4 . and j’s. For instance, here 5 and l are equal, but 5 an a ‘ # 511 0 T ; [5H 0] b _ 521 3'22 i "'1‘ 333 S 7 s” 0 T" : [S11 ll ] _ 3'21 322 . () .s S ¢ T“. [0" -"21 7* 0 [~/ I- MATRIX OPERATIONS ) )l . I ddd {WU 0r ”TOTC ”Id“ lLCS, ”le ”lust he ()1 the same dlmCllsIC 1] 0r ()(lCl 0 Int. l qu d t l H: lc‘ulllll lIldIlIX l5 f the bdlllC OldLl, Wllll CdLll ClC ll 1 l l the l 5 g 0 . Sun] 0‘ [he LOIICspondlng LlLlllCIlls l” Cdbll 0‘ the mdtllLCb dddbd lOgCIIlCl. [311 :81.) I: [I 0] B : l/izi B’Zl, 0 l a,H m2] .821 [£33 + 1 For instance, B+I=[ MAi'iux OPERATIONS 451 Two useful properties of matrix ad dition, [or any matrices S, T, and U of the same order, are as follows: LS+T=T+S ZJS+D+U:S+U+U) Two matrices can be multiplied only when the number of columns in the first matrix equals the number of rows in the second matrix. If this condition is met the matrices are said to be conformable with respect to multiplication. If the first matrix has order a X b and the second matrix has order I) X c, then the resulting product of these two matrices is a matrix of order a X 1'. The I] e. elements of the 1th row of the first S T [lit A'12 ‘ Ali/v] ’11 [h “‘TT5“ I 7 I, ‘21 522 ‘ 3'1}: i" 7‘ I I I [l [ . , [)2 I) ul Aill suh L U “11 “12 ”it ”21 “22 ”2t llul ”(12 “at es T, labeled U, start with the first row of S, which is enclosed in a box. Each element in this row is multiplied by the corresponding element in the first column of " l, which is also enclosed in a box. The sum of these b products equals u“ of U. ln other words “11 : Sii’ii T 512’21 + +-"ih’hi . This may be stated more generally as I) “1/ 2 Z SIAIA/ A—' for each element of Us MATRIX ALUEBKA xuvrtw 452 For example, 0 .312 0 m B = .821 0 323 , 'l : "2 0 0 0 713 3 X 3 3 X1 312712 Bn = [321711 + 323713 0 3 X l (The term Bn appears in the latent variable model.) K _ S T .nd U Some properties of matrix multiplication for any matrices , , a tha‘ are conformable are as follows: 1. ST at TS (except in special cases). 2. (.LT)U = S(TU). 3. S(T + U) 2 ST + SU. 4. ((5 + T) = ('S + (T (where c is a scalar). These properties come into play at many points throughout the book. T2: order in which matrices are multiplied is importantd‘l‘or 1h]; :ieals:or I i I I r u x - s C ‘ ’ ' ' 4 ' [rout/an by a matrix are istingu1s remu/(l [nation and postmulup . . . I‘) t ncepif U -- ST we can say that U results from the premultiphcation of ms a ' , — , ,. " ‘ ‘ ' tionobey l. l b b or the postmultiphca _ . , ‘ . e The transpose of a matrix interchanges its rows and colltlimns. Title . . . , . m transpose of a matrix is indicated by a prime () symbol fo ow g matrix. As an example, consider I and F : ’ ‘ ‘ " second The first row of I‘ is the first column of F , the second row fislthe 3 X 2 column and the third row is the third column. The order 0 dis mm); 1 ereas the order of I" is 7 X 3. The transpose of an a x b or cr n W) ‘ - leads to a b X a matrix. MA l'RIX OPERATIONS 453 Some useful properties of the transpose operator are listed below: I. (S’)’ = S. 2. (S + T)’ = S’ + T’ (where S and T have the same order). 3. (ST)’ = T’S’ (where matrices are conformable for mu] tiplication). 4. (STU)’ = U’T’S’ (where matrices are conformable for multiplication). Some additional matrix types and matrix operations are important for square matrices. A square matrix is a matrix that has the same number of rows and columns. An example of a square matrix is Yit 0 713 F = 0 Y22 0 0 Y32 733 The dimension of the F matrix is 3 X 3. The trace is defined for a square matrix. It is the sum of the elements on the main diagonal. For an n X :1 matrix S, tr(S) = Z s” l-l Properties of the trace include: I. tr(S) : tr(S’). 2. tr(ST) = tr(TS) (if T and S conform for multiplication). 3. tr(S + T) = tr(S) + tr(T) (if S and T conform for addition). The trace appears in the fitting functions and indices of goodness of fit for many structural equation techniques. If all the elements above (or below) are zero, the matrix is triangular. For models (which I discuss in Chapter 4) The B matrix contains the coefiicie latent variables on one another. To il the main diagonal of a square matrix instance, the B matrix for recursive may be written as a triangular matrix. nts of the effects of the endogenous lustrate, one such B matrix is 0 0 0 B : [321 0 0 (in .832 0 454 MATRIX ALUEBRA REVIEW Note that in this case the main diagonal elements are zero. However, triangular matrices may have nonzero entries in the main diagonal. A diagonal matrix is a square matrix that has some nonzero elements along the main diagonal and zeros elsewhere. For instance, 85, the covari- ance matrix (see Chapter 2), of the errors of measurement for the x variables, commonly is assumed to be diagonal. For 8,, 82, and 83, the population covariance matrix 85 might look as follows: VAR(5,) 0 0 o, = 0 VAR(82) 0 0 0 VAR(83) The zeros above and below the main diagonal represent the assumption that the errors of measurement for different variables are uncorrelated. A symmetric" matrix is a square matrix that equals its transpose (e.g., S = S’). The typical correlation and covariance matrices are symmetric since the 1'] element equals the ji element. For instance, VARUl) (‘oV(x,,.x2) covti‘hh)’ 2 = C()V(.\‘2,x,) VAlUxZ) (‘()V(.x‘l,x3) (‘()V(x3,xl) C()V(x3,x2) VAR(x3) For all the variables, the covariance of x, and x, equals the covariance of x]- and x,. Sometimes symmetric matrices, such as E, are written with blanks above the main diagonal because these terms are redundant. An identity matrix, I, is a square matrix with ones down the main diagonal and zeros elsewhere. The 3 X 3 identity matrix is l () 0 I 2 U l (l 0 (l 1 Properties of the identity matrix, I, include the following: 1. IS = SI : S (for any I and S conformable for multiplication). MATRIX OPhRA'I‘IONS 4 55 A vector that consists of all ones is a unit vector' UnthCClOI‘ products have some interesting properties. If you premultiply a matrix by a conformable unit vector, the result is a row vector whose elements are the column sums of the matrix. For example 4 1’T=[1 1 1}0 1:“, 2] 2 Postmultiplying ' ' ' ' a matrix by a conforming ' unit vector le‘ ' ' vector of row sums: dds [0 d Wlumn T1 ll Nob 1 _ f—1 ._‘._‘ I—J ll k» F T . . . Lmutually, if we both premultiply and postmultiply a matrix by conforming vectors, a scalar that equals the sum of all the matrix elements results 1"“ [ 4 ‘1 1 = 1 1 1 0 : l 2 i [I] [8] Using unit vectors and some of the other matrix properties we can compute the covariance matrix. Consider X an N X 1) matrix of NiobserVi [1011‘ ' " ‘ I‘— s for p variables. The 1 X p row vector of means for X is formed as ll)” The .1 . . . COlundeViatton- fort? of X requtres subtracting from X a matrix whose is consist o N X l vectors of th . ‘ e means for the corres d' v. , ‘ ‘ ‘ pon in f1:Sables :11 X. So every element in the first column equals the mean of mi ' v; ‘ ‘ ma e in X, every element in the second column equals the mean of 456 MATRIX ALGEBRA REVIhW MATRIX ()PERA'l‘lONS 457 ‘ , i, .This matrix Of means is The last line is the unbiased sample covariance matrix and contains the the second column Of X, ‘md so on variances of the variables in the main diagonal and the covariances of all I i I’X pairs of variables in the off-diagonal elements. All covariance matrices are N square and symmetric. h' l hen subtracted from X forms deviation from the mean scores: Suppose that 1 form a diagonal matrix from the main diagonal of S: w 1c 1, w ’ X— 1(L)I’X var(xl) 0 0 N D = 0 var(x2) 0 If the preceding deviation score matrix is represented by Z, then the P X P 0 0 V3r(X3) unbiased sample covariance matrix estimator 5 1s 1 where var(x,) represents the sample variance of x" The square root of D, S : (mil/Z represented as [)1/2, has standard deviations in its diagonal: N _ . .. 4 ’ 1/2 A numerical example illustrates these calculations. [var(xl)] 0 0 2 3 1 D“ = o [more 0 —l 1 1 0 0 [var()c3)]l/2 x z 0 4 2 _1 0 0 and [)"1/1 has the inverse of the standard deviations in its main diagonal. 2 3 1 If S is postmultiplied by 04/2, the first column is divided by the standard 1 1 deviation of xl, the second column is divided by the standard deviation of (LP/x : (1)“ 1 1 1 i _(l) 4 2 = [0 2 1] x2, and the third column is divided by the standard deviation of x}: N 4 $1 0 0 [var(x ”1/2 cov(xl, x2) cov(xl, X3) 1 1 2 1 2 l 0 2 i [var(x2)] / [var(x3)] / l 1 _ 0 2 cov(x,,x) cov x ,x .(‘)l’x= 1 to 2 11- 0 2 1 so \ want“ #342 N ] 0 2 l [var(x1)] [var(x3)] / 0 cov(,\’3,xl) c0v(x3,x2) 1/2 2 1 g 1/5 1/2 [mid/‘3” l *1 ‘1 0 [Var(x,)] [var(x2)] Z:X’l(*)llx: 0 2 1 _1 v2 *1 Premultiplying SD 1/2 by I) 1/2 leads to V'dri X1) COWX" X1) COW/‘1’“) 1 “0‘4le ‘2) COV(XIv 11) s -_ ( -——1——)Z'Z : (Kn/(X2, XI) var(,\‘2) COViXZv X3) [var(xl)var(,\2)]l/2 [var( xl)var(/\,)]l/2 ( N — 1) C()V(.\'3, X1) cowl-3» X2) Vaf(X3) l) 1/25” 1/2 = m, :(Wifil;:flcfi 1 ¥_COV(X2‘ A797,: ' [var(.x2)var(.il)] /’ [var(x2)var(.\3)]l/‘ l 6 5 1 cov(.r),x,) cov(x3,,xz) : — 5 10 4 < - 1/72, 1/2 1 3 ] 4 2 ““00”“ ‘1” [var(x‘)var(xl)] MA'I'RlX ALGEBRA RtiVll-LW 458 . . . . ‘. ,' onal The resulting matrix is the sample correlation matrix W-nathc'r‘ifl diaguus ' ' ‘ ' ' d x varia es. iese res elements equal to the correlations.ofltfliifIZICanariu/mc matrix S is p“:— and 4 " ' dimension matrix. 1 . _ . generdhli": liedngy 1) V2 where D ”2 is the diagonal matrix With the postmu ip 7 ' ‘ ' ' ' is the t dard deviations of x in its diagonal, then the resulting matrix s an ' . .. 1 le correlation matrix. . .~ ‘ .. .~ dwom. 5drThe nonnegative integer power of square mdlrlLCb occurs in thef “in“ a )sition of etl‘ects in path analysis. lt is defined as the number 0 P‘ ‘ g . p ‘ . ' . ‘ matrix is multiplied by itself. For instance, ' 0 0 all 0 a; 0 B12 , rim/321 1871 0 —. [ill 0 18M 0 O :Bllfill q , q ) I L l I (I (“I 5 UdlC “Hunt :3 d Stdldl Udnllly Clelb LJHLd ”TC aClLlnl H “I g) ,1 I S V i S l l .. . t ) h f » [prlCSCHtL d5 's I 0| (Cl . ll llC Ldb‘. 0 d ... X 2 [TTdUIX t C 0 7 dClCllllllldHl 15 = 511322 - 512571 If S is a 3 X 3 matrix, the determinant is , , . , s. ‘ . ' ' +5 .5, s“ 313.322.“ ‘ ‘ ' ' 5 5 5 533 ll -3. 321 5n 323 511322 33 12 2t l Siiszisn T Sit-“23532 ' d) f S increases, the formula for the determinant becomes Flor: A5 thc'or d 0 L i' a eneral rule for calculating determinants or complicamd: Thcic ‘5 )rdegr To explain this rule, the concepts of a riimor Square mam" 0f ddnil )(be defined. The minor of a matrix is the determinant mfmlil cafalt‘rliy onlftZined when the ith row and jth column of a matrix are 0 t e m removed, Consider the following matrix S. MATRIX OPERATIONS 459 The minor with respect to 5”, represented as |S”|, is ls l‘ 322 323 _ _ ll ‘ 332 333 ' 522533 523332 The minor of x22 is s s it 13 822i — 331 s33 : Sit-‘33 “ sllsll The cofactor of the element A“! is defined as (—1)”/ times the minor of 51/, If Clj=(—) 1'5 ill The cofaetors of each element of matrix S placed in the appropriate Ijth location creates a new matrix: +|Slli _. Slli +|SU| _|Szl| +lszzl —ISB| +|S3li ‘iS32l +|SB3| The determinant of a matrix can be found by multiplying each element in any given row (column) in the S matrix by the corresponding cofactor in the preceding matrix. This is then summed over all elements in the row (column). For example, if we do this for the first row of S, we obtain SiilSiil _ 312lSizl + SIJiSUi : 311(322533 “ 323332) *512(32i5n _ 323531) +513(321532 _ 322311) = 511322333 _ 311323532 Tsizszisn + 312523351 +313321332 “ 313322331 2 311322333 — 312321333 +3'i232353i — 313322331 +‘S‘13321332 - 311323332 Note that this formula is identical to the earlier formula for the determinant 460 MATRIX ALUEBRA REVIEW of a 3 X 3 matrix. Slightly dill'erent arrangements of terms occur depending on the row or column selected for expansion. However, regardless of which formula is chosen, the determinant will be the same. Useful properties of the determinant for square and conformable S and T and a scalar c include: I. |S’| z |S|. 2. If S = CT, then |S| = e"|T| (where q is the order of S). 3. If S has two identical rows (or columns), |S| = 0. 4. [ST] = |S||T|. The determinant appears in the fitting functions for the estimators of structural equations. It also is useful in finding the rank and inverse of matrices. The inverse of a square matrix S is that matrix S that, when S is pre- or postmultiplied by S 1, produces the identity matrix, I: SS 1 = S lS = I The inverse of a matrix is calculated from the adjoint and the determinant of a matrix. The adjoint of a matrix is the transpose of the matrix of cofactors defined earlier. Using the 3 X 3 S matrix, the adjoint of S is I +lsiil Tlszil + Slli adjS= _|SIZI +iszzl 'lsszl +isnl T Sui +isnl The inverse matrix is 1 S I: —(adjS) |S| To illustrate the calculation of the inverse, consider the simple case of a two-variable covariance matrix: S [20 10 10 20 ISI = (20)(20) ‘ (10)(10) = 400 ~ 100 = 300 . 20 , 10 Matrix of cofactors of S = —10 20 use 20 #10 “1‘ ‘ —10 20 MATRIX OPERATIONS 5-1:; 20 —10 300 —10 20 MUIleIy|ng S ’1 b - ' ‘ ySyieldsazxgd . Note that the inverse S ”Id 1 entity mamX. determinant, it is called a 3. are the following: 1. (S’)”=(S ‘). 2. (ST) l=T ls ‘;(STU)"'=U"T“S“ In manipulating the latent variable e inverses. In addition the inverse ap functions and in several other topics. Another important pro quations, we sometimes need to take pears m explanations of the fitting I. rank(S) g min(u, b), where u i' of columns. 2. rank(ST) s min[rank(S), rank(T)]. 1c roots. (Often such an equation is om this practice so that k is not ch use the same symbol.) represented as Ax = Xx. I depart fr confused with the factor loadings whi MATRIX ALUI-JIRA Rlelliw 462 The preceding equation may be rewritten as Su — eu = 0 (S ~ eI)u : 0 Only if (S ~ eI) is singular, does a nontrivial solution1 for this equation exist. If (S — e1) is singular, then [S .4 all = 0 Solving this equation for e provides the eigenvalues. . ! ix. To illustrate, suppose that S is a 2 X 2 correlation ma r . , 1.00 0.50 z 0.50 1.00 The (S 7 61) matrix is 1.00 ~ 6 0.50 S ' 61: 0.50 1.00 _ e The determinant is is 7 el] 2 (1.00 , e)2 — 0.25 = e2 , 243 + 0.75 5 [(1 t llvdlUCS [0| ””5 2 X 2 ll“: [W0 1.5 dnd 0.5, drC th Clgc \ Solution ) , 0 0 (.Ollcldllon llldIHX. Ld‘ah Clgcllvdluc, 6‘ lldb (1 Set 0i ClgLTlVLLl '5, u, (.155 (al‘ dlcd Wllll ll [0] delllplL, ”lb 6’ 0‘ 1.5 lCddb [0 ”TC iOIlUWIllg (Siel)u=0 1.00 — e 0.50 110] Z 0 0.50 1.00 — e u2 0.50 #050 M2 ,, 0.50111 + 0.50141 2 (l r050 °-”H‘“l=0 0.5011l -» 0.50112 2 0 "~ ' * that u is a nonzero ld xist if u = 0, As specmed hue, I assume lA trivial solution for e wou e vector. MATRIX optakA'i'ioNs 463 From this you can see that uJ = u2 and that an infinite set of values would work as the eigenvector for the eigenvalue of 1.5. Though the eigenvalues for the preceding example and for all real symmetri“ matrices are real numbers, this need not be true for nonsymmet- ric matrices. When the eigenvalue is a com lex number, say 2 = a + 1b, where a and b are real constants and i = v —1 , we commonly refer to the modulus or norm of 2 which is (a2 + lazy/2 Some useful properties of the eigenvalues for a symmetric or nonsym- metric square matrix S are l. A I) X 1) matrix S has b eigenvalues (some may take the same value). 2. The product of all eigenvalues for S equals IS]. 3. The number of nonzero eigenvalues of S equals the rank of S. 4. The sum of the eigenvalues of S equals the US. Eigenvalue and eigenvectors play a large role in traditional factor analyses. In this book they are useful in the decomposition of etl‘ects in path analysis discussed in Chapter 8. Quadrant" forms are represented by x’Sx (l x b)(b >< b)(b x1) which equals 464 MATRIX ALowitA RliVlliW Occasionally, structural equation models analyzed with the LISREL program (J‘oreskog and Sbrbom 1984) may report that a matrix is not positive-definite. For instance, suppose that we analyze the following sample covariance matrix S: 7 3 4 3 2 1 4 1 3 S is not positive-definite, since x’Sx is zero for some x #= O (e.g., x’ = [l —1 ~ 1]). Indeed, S is singular (|S| = 0), and singular matrices are not positive-definite. Consider the following three matrices: i? ll l‘i llsl l3 3] Assume that these are estimates of the covariance matrix of the distur- bances from two equations. None is positive-definite. The failure of the first two to be positive-definite indicates a problem. In the first case the covariance (: 3) and variances (2 and 1) imply an impossible correlation value ( = 3/ J27) The middle matrix has an impossible negative disturbance variance (= ~ 2). Whether the nonpositive detinite nature of the last matrix is troublesome depends on whether the variance of the first disturbance should be zero. Identity relations (e.g., 7h = 112 + 7h) or when measurement error is absent (e.g., x, 3&1) are two situations where zero disturbance variances make sense. However, when zero is not a plausible value, then the analyst must determine the source of this improbable value The vec operator is the operation of forming a vector from a matrix by stacking each column of a matrix one under the other. For instance: 0 vecB ve[0 [312] 321 t = C : fill 1 1812 l The vec operator appears in Chapter 8‘ MATRIX omaaA'riONs 465 A Kronetker product (or a direct TU" X '1), is defined as product) or two matrices, 5(1) X 4) and 311T - - - squ S o T = ' splT - - - SMT Each element of the left ' ' n matrix, S, ' ‘ ' 3”]- A” of these wbmdlr. \ is multiplied by T to form a submatrix example is: l [711 Y121® .812 2 Y“ 711312 Yiz 717.8” [321 1 Y B i ‘1 21 711 YlZBZl Yiz Kronecker’s products appear in the formul errors of indirect elTects in Chapter 8 35 for the asymptotic Standard ...
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