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MIT 12 215 - Study Notes

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12.215 Modern NavigationToday’s ClassBasic vectors and matricesLengths and dot productsAngles between vectorsPlanesMatrices and linear equationsSolving linear equationsRules of matrix multiplicationFactorizationCharacteristics of LUTransposeMatrix rankVector spacesThe null spaceEigenvectors and EigenvaluesEquation for eigenvaluesDiagonalization of a matrixOther matrices that we will encounterRotation matricesDirection cosines relationshipSmall angle rotationsSummary12.215 Modern NavigationThomas Herring ([email protected]), http://geoweb.mit.edu/~tah/12.21510/14/2009 12.215 Modern Naviation L09 3Today’s Class•Review of linear Algebra. Class will be based on the book “Linear Algebra, Geodesy, and GPS”, G. Strang and K. Borre, Wellesley-Cambridge Press, Wellesley, MA, pp. 624, 1997•Topics to be covered will be those later in the course•General areas are:–Vectors and matrices–Solving linear equations–Vector Spaces–Eigenvectors and values–Rotation matrices10/14/2009 12.215 Modern Naviation L09 4Basic vectors and matrices•Important basic concepts•Vectors: A column representing a set of n-quantities–In two and three dimensions these can be visualized as arrows between points with different coordinates with the vector itself usually having on end at the origin–The same concept can be applied to any n-dimensional vector–Vectors can be added and subtracted (head-to-tail) by adding and subtracting the individual components of the vectors.–Linear combinations of vectors can be formed by scaling and addition. The result is another vector e.g., cv+dw–(Often a bold symbol will be used to denote a vector and some times a line is drawn over the top).10/14/2009 12.215 Modern Naviation L09 5Lengths and dot products•The dot product or inner product of two vectors is defined as:•The order of the dot product makes no difference.•The length or norm of a vector is the square-root of the dot product•A unit vector is one with unit length •If the dot product of two vectors is zero they are said to be orthogonal (orthonormal if they are unit vectors)•The components of a 2-D unit vector are cos and sin of the angle the vector makes to the x-axis. € v ⋅w = v1w1+ v2w2+L + vnwn10/14/2009 12.215 Modern Naviation L09 6Angles between vectors•The cosine formula:•Schwarz inequality: The dot product of any two vectors is less or equal the product of the lengths of the two vectors•The representation of a plane is by the vector that is normal to it. For a plane through the origin, all vectors in that plane must have zero dot product with the normal. This then provides an equation for the plane.€ cosθ =v ⋅w| v || w |10/14/2009 12.215 Modern Naviation L09 7Planes•If a plane does not contain the origin, then the coordinates of a point on the normal containing the plane specifies the plane. The dot product of the normal and points in the plane then is a constant.•If the normal to the plane is a expressed as a unit vector, the constant is the closest distance of the plane to the origin.•In N-dimensional space, the concept of a plane is the same: it is defined by n.v=d•Any two non-collinear vectors define a plane10/14/2009 12.215 Modern Naviation L09 8Matrices and linear equations•Any set of linear equations (i.e., equations which do not contain powers or products of the unknowns) can be written in matrix form with the coefficients of the linear equations being the elements of the matrix.•The rows and columns of matrices are themselves vectors.•A matrix represents a linear combination of the elements of a vector to form another vector of possibly different length:€ Ax = b where x and b are vectors of length n and mA is a m - rows and n column matrix10/14/2009 12.215 Modern Naviation L09 9Solving linear equations•If the x and b vectors are the same length, then given A and b it is often possible to find x (sometimes this is not possible and sometimes x may not be unique).•There a many methods for solving this type of system but the earliest ones are by elimination i.e., linear combinations are formed of the rows of the matrix A that eliminate one of the elements of x. The process is repeated until only one element of x remains (which is easily solved). Back substitution allows all the components of x to be computed.•This process is sometimes viewed as multiplying by eliminator matrices.10/14/2009 12.215 Modern Naviation L09 10Rules of matrix multiplication•The product of two matrices A (n-rows and m-columns) by B (r-rows and c-columns) is only possible if m=r•The resultant matrix has n-rows and c-columns.•In general, AB does not equal BA even the matrices are square•A matrix multiplication is the dot products of rows of the first matrix with the columns of the second matrix•Matrix multiplication is associative (AB)C=A(BC) but not commutative•A matrix is invertible if A-1 such that A-1A=I where I is a unit matrix, exists.10/14/2009 12.215 Modern Naviation L09 11Factorization•In factorization a matrix A is written as A=LU where is L is a lower triangular matrix and U is an upper triangular matrix.•The individual matrices L and U are not unique (L can be multiplied by a scalar and U divided by the same scalar with out changing the product. Convention has the diagonal of L being 1’s.•Why factorize? Since forms are lower triangular, substitute down (L) and up (U) the matrix€ Solve Lc = b then solve Ux = c to solve Ax = b10/14/2009 12.215 Modern Naviation L09 12Characteristics of LU•When the rows of A start with zero so do the corresponding rows of L; when the columns of A start with 0 so do the columns of U.•Many estimation problems are “band-limited” i.e., only a small number of the elements around the diagonal are non-zero; the L and U matrices will also be band-limited but the inverse of such a matrix is normally full. (Factorization saves time and space).•http://web.mit.edu/18.06/www/Course-Info/Mfiles/slu.m is a link to an SLU matlab code (also code at same site that pivots the matrix which is a more stable approach).10/14/2009 12.215 Modern Naviation L09 13Transpose•The transpose of a matrix is the matrix with rows and columns switched. Usually denoted as AT or sometimes A’•Some rules: (AB)T=BTAT• A symmetric matrix is one for which A=AT•The products ATA and AAT generate symmetric matrices. We will see these forms many times in when we cover estimation and statistics.10/14/2009 12.215 Modern


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