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A matrix A Rn m is a tabular arrangement of real numbers with n rows and m columns An example for n = 2 and m = 3 is A= 020 000 314 171 000 200

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Sometimes it is necessary to name the elements of a matrix individually In this case, we state that A = [ai,j ] for i = [1, n] and j = [1, m] For example in the above example, a2,3 = 200 For matrices A and B Rn m , the notation A = B means that ai,j = bi,j for all i [1, n] and j [1, m] The null matrix Zn m = 0 Rn m has zi,j = 0 for all i [1, n] and j [1, m] The identity matrix A = In Rn n has ai,j = 1 0 for i = j otherwise

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1 0 Normally the subscripts on I, Z, and 0 will 0 1 be dropped, because the dimensions can be determined directly based on the context For example, I2 = 459

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Copyright 2008 by The McGraw-Hill Companies Click here for terms of use

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APPENDIX B LINEAR ALGEBRA REVIEW

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When a matrix has the same number of rows and columns (ie, n = m), the matrix is said to be square The elements ai,i for are referred to as the diagonal elements of A (ie, diag(A) = [a11 , , an,n ] A square matrix A is called diagonal when aij = 0 for i = j The notation A = diag([a1 , , an ]) will be used to state that A is a diagonal n n matrix with main diagonal elements speci ed by the vector [a1 , , an ] The identity matrix is an example of a diagonal matrix A square matrix A is symmetric if and only if it is square and aij = aji for all i, j [1, n] A square matrix A is skew-symmetric if and only if aij = aji for all i, j [1, n] This implies that a skew-symmetric matrix has diagonal elements all equal to zero Skew symmetric matrices are useful for representing the cross product as a matrix operation, see eqn (B14) There are various special classes of matrices One type of matrix is a permutation matrix A matrix P is a permutation matrix when each row and column of P contains exactly one 1 and all other elements of P are 0 The term vector can have both physical and algebraic meanings For example, the line segment from point P1 to point P2 can be represented as the vector d = P2 P1 Note that this vector is well-de ned without any discussion of reference frames Also, the velocity vector v of a point has well-de ned physical interpretation The velocity vector points in the direction of travel and has magnitude equal to the speed of the point If a frame-of-reference is speci ed, then either d or v can be represented algebraically as a special matrix that only has one column When represented with respect to a speci c reference frame, the i-th element of a vector v is the projection of v onto the i-th coordinate axis of the reference frame Alternatively, for convenience, data or variables are sometimes organized into a column matrix (or vector) without the vector having a clear physical interpretation with respect to any single frame of reference An example of this is the vector of Euler angles and the vector of Euler angle derivatives discussed in Section 2721 When we want to refer to the component in the i-th row of the vector v, we use the notation vi instead of vi,1 Throughout the majority of this text all vectors will be represented as column vectors The only time that

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