University of Ferrr Stefno Bonnini Multivrite problems nd mtrix lgebr
Multivrite problems Multivrite sttisticl nlysis dels with dt contining observtions on two or more chrcteristics (vribles) ech mesured on set of objects (sttisticl units) Exmple : exmintion mrks, bout courses (Mechnics, Vectors, lgebr, nlysis, Sttistics), chieved by 88 students Exmple : weights of cork deposites (centigrms) for 8 trees in the four directions (N, E, S, W) Exmple : flower mesurements (sepl length, sepl width, petl length, petl width) on flowers belonging to certein species of iris
Multivrite problems n sttisticl units nk pieces of informtion k vribles vilble informtion Dtset n k mtrix Exmple: dt mtrix with students where X ge in yers t entry to university, X mrks out of in n exmintion t the end of the first yer nd X sex Vribles units X X X 8 8 89 8 8 9
Multivrite problems Some multivrite problems: Exmple : study how the mrk in the exmintion of «Sttistics» (dependent vrible) is ffected by or cn be predicted s function of the mrks in other exmintions or other vribles such s ge, sex, etc (explntory vribles) regression problem Exmple : study how to combine the informtion on the performnce of the students on the exmintions to determine the globl performnce of ech student with just one, or two or less thn vlues fctor nlysis, principl component nlysis, composite indictor Exmple : study how to group students with similr performnces by considering the whole set of exmintions cluster nlysis
Multivrite problems The generl n k mtrix which represents dtset with n sttisticl units nd k vribles cn be written s follows: Vribles Units X X v X k x x v x k u x u x uv x uk n x n x nv x nk This mtrix cn be denoted X or (x uv ) x u xu x uv xuk x ( v) x v x uv xnv
m n mtrix is tble with m rows nd n columns: 9 8 8 In this cse the mtrix hs rows nd columns If mn then it is clled squre mtrix ij i denotes the row j denotes the column
mtrix with dimension n is clled row vector: ( ) mtrix with dimension m is clled column vector or simply vector: c unit vector is vector of ones:
Given the mtrices nd B, their sum is defined s C B, where c b ij Exmple: ij ij 9 8 8, B 9 9 9 B C 8 8 8 b c 8
The product of m n mtrix nd sclr (single vlue) λ is clled sclr multipliction nd it consists in mtrix with the sme dimension of, obtined by multiplying ech element of by λ Cλ c λ ij ij Exmple: λ 9 8 8 8 8 8 9
B B C ) ( ' j i ij c b
where the elements of C re equl to: c c c c c c C
Note tht: B C ( m n) ( n h) ( m h) Thus the product between row vector nd column vector is sclr; the product between column vector nd row vector is mtrix: b c n n b C n n n n
Exmples: ( ) b 8 b b ( ) 8 b
The trnspose of the mtrix ( ij ) is the mtrix ( ji ) whose rows correspond to the columns of : Exmple: 9 8 9 8 The squre mtrix ( ij ) is symmetric if ij ji or equivlently if Exmple: 9 9 8
null mtrix is mtrix with ll elements equl to digonl mtrix is squre mtrix whose elements not in the min digonl re ll equl to ( ) n n dig,, L
The trnspose stisfies the following properties: ( ) (B) B (B) B digonl mtrix is squre mtrix whose elements not in the min digonl re ll equl to
The trce of ( ij ) is the sum of the elements in the min digonl of : Exmple: tr()σ i ii 8 tr 9 ( ) Min digonl
The trce stisfies the following properties for (m m), B (m m), C (m n), D (n m) nd sclr λ: tr(λ) λ tr()tr( ) tr(b)tr()tr(b) tr(cd)tr(dc)σ i,j c ij d ji tr(cc )tr(c C)Σ i,j c ij 8
Given the mtrix The determinnt of is det ( ) 9
Given the m m mtrix The determinnt of is det m ( ) ij ij j m i ij where the cofctor ij is the product of (-) ij nd the determinnt of the mtrix obtined fter deleting ith row nd jth column of (minor) ij for ny i,j Cse m: det( ) ( ) ( ) ( )
Computtion of the determinnt of rd order mtrix (Srrus rule): det ( ) ( )
det() ( ) Exmple: ( ) ( ) ( ) ( ) 8 det or lterntively:
Properties of the determinnt If dig(,, n ) then det() n Π i i det(λ) λ λ n det(b) B B If hs two equl rows or two equl columns then det() If hs row of zeros or column of zeros then det() If B is the mtrix obtined exchnging the position of two rows or two columns of then det(b) -det() det()det( ) 8 If B is the mtrix obtined by summing to row or column of liner combintion of the other rows or columns of respectively then det(b)det() 9 squre mtrix is non-singulr if det() ; otherwise is singulr
Exmple: B 9 ) det( 9 ) det( B 9) ( 9 ) det( ) det( B 9 B 9 ) det( B
The inverse of the squre mtrix is the unique mtrix - stisfying: I dig() Identity mtrix I The inverse - exists if nd only if is non singulr, tht is, if nd only if det()
The identity mtrix is digonl mtrix where ll the elements in the min digonl re equl to I Properties of I I I I I
( ) 9 9 88 9 ) ( 9 B B 9 9 88 9 B
Exmple : Let us consider the following system of equtions The solution is 8
squre mtrix is orthogonl if I The following properties hold: - I ± i j, i j; i i, i; (i) (j), i j; (i) (i), i; Exmple: 9
Vectors x,,x k re clled linerly dependent if there exist numbers λ,, λ K not ll zero such tht λ x λ K x k Otherwise the k vectors re linerly independent Let W be subspce of R n Then bsis of W is mximl linerly independent set of vectors Every bsis of W contins the sme (finite) number of elements This number is the dimension of W If x,,x k is bsis for W then every element x in W cn be expressed s liner combintion of x,,x k Exmple: The dimension of WR is bsis for R is x (,,), x (,,) nd x (,,) s mtter of fct x, x nd x re linerly independent nd every vector (,, ) cn be expressed s liner combintion of x, x nd x : x x x
The rnk of n k mtrix is defined s the mximum number of linerly independent columns (rows) in The following properties hold for the rnk of, denoted with r(): r() is the lrgest order of those (squre) submtrices of with non null determinnts r() min(n,k) r()r( ) r( ) r( ) r() If nk then r()k if nd only if is non-singulr Exmple: det 9 Thus the r() ( )
If is squre mtrix of order n, in some problems we re interested in finding vector x nd sclr λ which stisfy the following property: eigenvlue λ λ ( I) x x x eigenvector trivil solution is x, ny λ R
The n eigenvlues of λ,, λ n re the n solutions of the chrcteristic eqution λi Properties of the eigenvlues of : Π i λ i tr() Σ i λ i r() equls the number of non-zero eigenvlues The set of ll eigenvectors for n eigenvlue λ i is clled the eigenspce of for λ i ny symmetric n n mtrix cn be written s ΓΛΓ Σ i λ i γ (i) γ (i) where Λ is digonl mtrix of eigenvlues of nd Γ is n orthogonl mtrix whose columns re eigenvectors with γ ((i) γ (i)
Exmple: The chrcteristic eqution is: λ λ λ By computing the determinnt we hve: ( ) λ λ λ λ λ The solutions represent the eigenvlues of : λ λ λ ( )[ ] ( )( )( ) 9 λ λ λ λ λ λ λ
The eigenvlue with mximum bsolute vlue λ is clled dominnt There is n infinite number of eigenvectors x which stisfy (-I)x x x x