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# fa11_mae260_linalg3 - MAE260 Lecture 8 Linear Algebra Part...

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MAE260 Lecture 8 Linear Algebra: Part I November 24, 2011

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Fall 2011 MAE260 8–1 Matrices Some notations: – v R n : v is a n -dimensional vector whose entries are all real. A R n × m : A is a m by n matrix whose entries are all real We can also define vectors and matrices in complex domain (and the textbook is written in this domain), but for now we will focus on real vectors and matrices for which we can easily talk about their geometric meanings. I assume that everyone is familiar with matrix addition and multipli- cation – so, I will skip them. Your probably first encounter of matrices in the context of System of Linear Equations: Example: Dad gives 1/10 of his income to his 1st kid and 1/6 to his 2nd kid, while Mon gives 1/9 of her income to the 1st kid and 1/8 to the 2nd one. Dad’s income = x D , Mom’s income = x M , 1st kid’s allowance = y 1 , and 2nd kid’s allowance = y 2 . Then, we have y 1 = 1 10 x D + 1 6 x M y 2 = 1 9 x D + 1 8 x M , which can be written as: y 1 y 2 = 1 10 1 6 1 9 1 8 x D x M Two types of problems can be paused with the above representa- tion. Forward problem: for given x D and x M , find y 1 and y 2 simply matrix multiplication November 24, 2011
Fall 2011 MAE260 8–2 Backward problem: for given y 1 and y 2 , what is appropriate x D and x M ? linear equation solving, which have interested many people and is still a research problem. We will revisit linear equation solving in the later part. Another encounter of matrices: coordinate transformation and/or vector rotation. Convince yourself that: if you rotate a 2-dimensional vector v = v x v y by θ around z -axis, i.e., the direction of i × j where i and j are the unit vectors in x and y direction, respectively, then resulting vector v = v x v y can be obtained by v = P θ v where P θ = cos θ - sin θ sin θ cos θ . If you further rotate v by φ around z axis to get resulting vector ̃ v , convince yourself that: ̃ v = P φ v = P φ ( P θ v ) = ( P φ P θ ) v , which further gives: P φ + θ = P φ P θ Note the difference between P θ in the above and R θ in (8.1.25) in the textbook. What causes this difference? November 24, 2011

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Fall 2011 MAE260 8–3 Matrix Inversion We define A - 1 , the inverse of a matrix A R n × n as matrix with the property A - 1 A = I where I R n × n is the identity matrix whose diagonal entries are all one and off-diagonal entries are all zero. Note that if A - 1 A = I then AA - 1 = I , i.e., A is the inverse matrix of its inverse matrix A - 1 . Also, the inverse of the product is the product of the inverses in reverse: ( AB ) - 1 = B - 1 A - 1 Inverse matrix does not necessarily exist. First, it is defined only for a square matrix. Second, it exists only for a non-singular matrix, which is equivalent to a matrix with non-zero determinant. One general way to calculate the inverse of a square matrix A R n × n , if it exists, is to use the following expression: A - 1 = A T C A where ( ) T and denote the transpose and the determinant of a matrix, respectively. The cofactor matrix of A C is defined as: ( A C ) ij = ( - 1 ) i + j [
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fa11_mae260_linalg3 - MAE260 Lecture 8 Linear Algebra Part...

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