L93 - Fall 2003 Math 308/501502 9 Matrix Methods for Linear...

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Fall 2003 Math 308/501–502 9 Matrix Methods for Linear Systems 9.3 Matrices and Vectors Mon, 10/Nov c ± 2003, Art Belmonte Summary (Read Chapter 11 of your lab manual for additional material.) Types of matrices An m × n matrix is a rectangular collection of elements arranged in m rows and n columns. These elements are real or complex numbers or symbolic expressions. We’ll denote matrices with uppercase boldface roman letters; e.g., A , B , etc. Thus A = ± a ij ² is a matrix whose -th entry is a .A row vector is a 1 × n matrix, whereas a column vector is an n × 1 matrix. Vectors will be signified by lowercase boldface roman letters; e.g., v , x ,etc.A matrix is square if m = n and diagonal if it is square with all off-diagonal entries being zeros. The elements of a zero matrix or a zero vector are all zeros. Either is denoted by 0 . The Algebra of matrices Matrix Addition and Scalar Multiplication Provided that matrices A and B have the same dimensions, their sum is given by A + B = ± a ² + ± b ² = ± a + b ² . Scalar multiplication is defined by r A = r ± a ² = ± ra ² scalar is a real or complex number or a symbolic expression. The expression - A stands for ( - 1 ) A and matrix subtraction is defined by A - B = A + ( - 1 ) B . Here are some properties of matrix addition and scalar multiplication. A + ( B + C ) = ( A + B ) + C A + B = B + A A + 0 = A A + ( - A ) = 0 r ( A + B ) = r A + r B ( r + s ) A = r A + s A r ( s A ) = ( rs ) A = s ( r A ) Matrix multiplication Let A = [ a ik ]bean m × n matrix and B = ± b kj ² an n × p matrix. The matrix product is defined by C = AB = A * B = ± c ² ,where c = n ³ k = 1 a b .Ino ther words, c is the dot product of the i th row of A with the j th column of B . (NOTE: This is different that the array product in MATLAB, defined by P . * Q = ± p ² . * ± q ² = ± p q ² ,for matrices P and Q having the same dimensions.) Here are some properties of matrix multiplication. They are respectively known as associativity, left distributivity, right distributivity, and another instance of associativity. ( AB ) C = A ( BC ) A ( B + C ) = AB + AC ( A + B ) C = + BC ( r A ) B = A ( r B ) NOTE what is missing: commutativity! In general, AB 6= BA , even if both these products are defined. (Indeed, one or both of them may not be defined.) Matrices as Linear Operators A consequence of the aforementioned properties of matrix multiplication is that an m × n matrix A defines a linear operator from R n to R m (or else C n to C m ): A ( x + y ) = Ax + Ay and A ( r x ) = r Ax .Fur thermore , AB constitutes a composition of the linear operators A and B and is itself a linear operator; i.e., AB = A B . Geometric examples of linear operations that act on vectors are stectching, contracting, rotation, and reflection.
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L93 - Fall 2003 Math 308/501502 9 Matrix Methods for Linear...

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