5-algebra_2_

Linear Algebra Vector:     1 T2 1 2 1 2 'n n n x x x x x x x x x x               - linear space nR 15-algebra(2) 2 A set of vectors is said to be linearly dependent if there exist real numbers , not all zeros, such that If (*) holds only for , then this set of vectors is said to be linearly independent.  mxxx ,, 21 )( .02211  mmxxx   m ,, 21 021  m  2 :R 1x 2x 21 xx  1x 2x 3x5-algebra(2) 3 In general, in the space, there are vectors that are linearly independent, but any vectors must be linear dependent. nR n 1n The dimension of a linear space is the maximum number of linearly independent vectors in the space. So, the space is an dimensional linear space. nR n 5-algebra(2) 4 A set of linearly independent vectors in is called a basis if every vector in can be expressed as a unique linear combination of the vectors in the set. nR nR Any set of linearly independent vectors is a basis for . n nR Proof: Let be linearly independent.  nqqq ,, 21 For any , are linearly dependent. nRx xqqq n ,,, 21  121 ,,, nn  There are , not all zero, s.t., 012211   xqqq nnn   5-algebra(2) 5 012211   xqqq nnn   Observe that . (otherwise, would be linearly dependent) 01 n nqqq ,, 21 n n n nn qqqx 1 2 1 2 1 1 1          Uniqueness? Let: nn nn qqq qqqx       2211 2211 nnn qqq )()()(0 222111    nn   ,, 2211? 5-algebra(2) 6 Determination of the coordinates ni i  ,,2,1,  nnqqqx   2211   xqqq n n                     2 1 21   xqqq n n 1 21 2 1                      5-algebra(2) 7 For an orthonormal basis,                                              1 0 0 ,, 0 1 0 , 0 0 1 21    n iii   xIxxiii n n                1 21 2 1      5-algebra(2) 8 Example: 2 3 1 Rx                     2 2 , 1 3 21 qq                                              2 1 3 1 4 3 4 1 2 1 2 1 3 1 21 23 1 2 1   21 2qqx  5-algebra(2) 9 Norms of vectors (length or magnitude) Any real-valued function of , , can be a norm if x x xx  ,01) and iff ; 0x 0x  Rxx   ,2) ; 212121 ,, xxxxxx 3) . 1x 2x 21 xx  Triangular inequality 5-algebra(2) 10 Example of norms: (1 norm) pn i p ip i i n i i n i i xx xx xxxxx xx 1 1 1 2 2 1 1 max :'                  (2 norm or Euclidean norm) (∞ norm) (p norm) 5-algebra(2) 11 Orthonormalization • is normalized if ; • and are orthogonal if 1' xxx 1x 2x 0' 21 xx A set of vectors is said to be orthonormal if ,,,2,1, mixi        ji ji xx ji 1 0 ' 5-algebra(2) 12 Schmidt orthonormalization procedure: Given linear independent,,,, 21 meee  1 1 111 , u u qeu  2 2 212122 ,)'( u u qqeqeu  1e 2e 1q 2u 3 3 3 1 3 1 2 3 2 3 3 ( ' ) ( ' ) , u u e q e q q e q q u     m m mmmmmmmm u u qqeqqeqqeqeu   ,)'()'()'( 112211  5-algebra(2) 13 Example:              2 1 , 1 1 21 ee                                 2 2 2 2 2 1 2 1 , 1 1 1 1 111 u u qeu 2 2 2 1 2 1 2 2 2 1 2 1 3 2 2 2 2 ( ' ) , 2 12 2 2 22 2 u u e q e q q u                                        5-algebra(2) m equations with n unknowns 14 11,,,   nmnm RxRyRAyAx Linear algebraic equations Let  naaaA 21 yAx  yxaxaxa nn  2211 is a linear combination of the columns of y A 5-algebra(2) 15 - Range space of : All possible linear combinations of columns of . The rank of is the dimension of the range space. A A A Rank( ) = the largest number of linearly independent columns of = the largest number of linearly independent rows of = size of the largest square submatrix whose determinant is non-zero A A A  min ,m n 5-algebra(2) 16 Example:  4321: 0202 4321 2110 aaaaA             1a and are linearly independent2a 24213 2, aaaaa  Any 3 columns are linearly dependent 2)(:)(Rank  AA  5-algebra(2) 17 - A vector is a null vector of if . x A 0Ax AThe null space of consists of all its null vectors. AA Nullity ( ) = maximum number of linear independent null vectors = number of columns of - rank( ) A            0202 4321 2110 A Nullity( ) = 4 - 2 = 2 A 5-algebra(2) 18            0202 4321 2110 A 0Ax                                 1 0 2 0 , 0 1 1 1 21 nn and are linearly independent and form a basis of the null space, that is: Null space = 1n 2n  Rnn  212211 ,:  5-algebra(2) 19 Theorem: Given nmRAyAx  , 1. Solution exists iff , that is, x range( )y A     yAA ,  2. Solution exists for any iffx y   mA  (full row rank) 5-algebra(2) 20 Theorem: Given . Assume that solution exists. Let . nmRAyAx  , px )(Ank  2. : then,0k 1. : solution is unique; 0k ,2211 kkp nnnxx    where is a basis of Null( ).A knnn ,,, 21  5-algebra(2) 21 - Determinant and inverse of a square matrix       1 1 , ,)(det j ijij n i ijij iCa jCaA ji nj ij j ij a a a C                   )1( 1   : Inverse of 1A A IAAAA   11 Inverse of exists iffA .0)(det A 5-algebra(2) 22   T 1 Adj( ) det( ) det( ) ijCA A A A    Example:                   1121 1222 12212211 1 2221 12111 1 aa aa aaaaaa aa A A shorthand method for computing 1A    1 4 3 4 1 2 1 2 1 10 01 4 2 4 2 4 3 4 1 01 10 10 4 3 4 1 21 10 10 31 21 40 10 01 21 23                                                   AI IA 5-algebra(2) 23 - Similarity transformation nRxAxx  ,Consider 0)(det,  QQzxLet (state transformation) zAAQzQAxQxQz   :111  AQQA 1 (similarity transformation) AA and are said to be similar (to each other) 5-algebra(2) 24 - Eigenvalues and eigenvectors x Ax x Ax 0,  xxAx  0,0)(  xxAI 0)(:)(det   AI Eigenvalues For a given eigenvalue : 0,  xxAx  Eigenvectors 5-algebra(2) 25 - Each eigenvalue corresponds to one or more linearly independent eigenvectors - Eigenvectors corresponding to different eigenvalues are linearly independent - If eigenvalues of are all distinct: A n ,,, 21  linearly independent eigenvectors:n nqqq ,,, 21  nnn qAqqAqqAq   ,,, 222111                   n nn qqqqqqA      2 1 2121 Q Q5-algebra(2) 26              1 2 1     QQA              n QAQ     2 1 1 Diagonalization! - If eigenvalues of are not all distinct: A Example: 0)()(det, 41 4   AIRA 1  (4 repeated eigenvalues) 5-algebra(2) 27 0)( 1  qAI Suppose or 3)( 1  AI 134)(Nullity 1  AI One (linearly independent) eigenvector, 1q 121 )( qqAI  2q 231 )( qqAI  3q 341 )( qqAI  4q are generalized eigenvectors, linearly independent, and: 4321 ,,, qqqq            4134 3123 2112 111 qqAq qqAq qqAq qAq     5-algebra(2) 28 4134 3123 2112 111 qqAq qqAq qqAq qAq                          1 1 1 1 43214321 1 1 1     qqqqqqqqA Q Q JQAQ               : 1 1 1 1 1 1 1 1     (Jordan block of size 4) 5-algebra(2) 29 0)( 1  qAI 0)()(det, 41 44    AIRA Suppose or 2)( 1  AI 224)(Nullity 1  AI Two linearly independent eigenvectors, 11, pq Two chains of generalized eigenvectors                  1 1 1 1 21212121 1 0 1     ppqqppqqA        2 11 J J QAQ 5-algebra(2) 30                  1 1 1 1 13211321 0 1 1     pqqqpqqqA        2 11 J J QAQ - Example (an application of Jordan block) JQAQ 1                QA Q QAQQAQJ detdet det 1 detdetdetdetdet 11       i n i JA    1 detdet A matrix is nonsingular iff it has no zero eigenvalue! Or: 5-algebra(2)