28-review2

28-review2 - CHECKLIST SECOND MIDTERM, Math21b, O.Knill The...

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Unformatted text preview: CHECKLIST SECOND MIDTERM, Math21b, O.Knill The EIGENVECTORS AND EIGENVALUES of a matrix A reveal the structure of A . Diagonalization in general eases the computations with A . It allows to find explicit formulas for LINEAR DYNAMICAL SYSTEMS x 7 Ax . Such systems are important for example in probability theory. The dot product leads to the notion of ORTHOGONALITY, allows measurements of angles and lengths and leads to geometric notations like rotation, reflection or projection in arbitrary dimensions. Least square solutions of Ax = b allow for example to solve fitting problems. DETERMINANTS of matrices appear in the definition of the characteristic polyomial and as volumes of parallelepipeds or as scaling values in change of variables. The notion allows to give explicit formulas for the inverse of a matrix or to solutions of Ax = b . ORTHOGONAL ~v ~w = 0. LENGTH || ~v || = ~v ~v . UNIT VECTOR ~v with || ~v || = ~v ~v = 1. ORTHOGONAL SET v 1 , . . . , v n : pairwise orthogonal. ORTHONORMAL SET orthogonal and length 1. ORTHONORMAL BASIS A basis which is orthonormal. ORTHOGONAL TO V v is orthogonal to V if v x = 0 for all x V . ORTHOGONAL COMPLEMENT OF V Linear space V = { v | v orthogonal to V } . PROJECTION ONTO V orth. basis v 1 , . . . , v n in V , perp V ( x ) = ( v 1 x ) v 1 + . . . + ( v n x ) v n . GRAMM-SCHMIDT Recursive u i = v i- proj V i- 1 v i , w i = u i / || u i || leads to orthonormal basis. QR-FACTORIZATION Q = [ w 1 w n ], R ii = u i , [ R ] ij = w i v j , j > i . TRANSPOSE [ A T ] ij = A ji . Transposition switches rows and columns. SYMMETRIC A T = A . SKEWSYMMETRIC A T =- A ( R = e A orthogonal: R T = e A T = e- A = R- 1 ). DOT PRODUCT AS MATRIX PRODUCT v w = v T w . ORTHOGONAL MATRIX Q T Q = 1. ORTHOGONAL PROJECTION onto V is AA T , colums ~v i are orthonormal basis in V ....
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