# PartII - Part II Linear Algebra c Copyright Todd Young and Martin Mohlenkamp Mathematics Department Ohio University 2007 Lecture 8 Matrices and

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Part II Linear Algebra c c Copyright, Todd Young and Martin Mohlenkamp, Mathematics Department, Ohio University, 2007

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Lecture 8 Matrices and Matrix Operations in Matlab Matrix operations Recall how to multiply a matrix A times a vector v : A v = p 1 2 3 4 Pp 1 2 P = p 1 · ( 1) + 2 · 2 3 · ( 1) + 4 · 2 P = p 3 5 P . This is a special case of matrix multiplication. To multiply two matrices, A and B you proceed as follows: AB = p 1 2 3 4 1 2 2 1 P = p 1 + 4 2 + 2 3 + 8 6 + 4 P = p 3 0 5 2 P . Here both A and B are 2 × 2 matrices. Matrices can be multiplied together in this way provided that the number of columns of A match the number of rows of B . We always list the size of a matrix by rows, then columns, so a 3 × 5 matrix would have 3 rows and 5 columns. So, if A is m × n and B is p × q , then we can multiply AB if and only if n = p . A column vector can be thought of as a p × 1 matrix and a row vector as a 1 × q matrix. Unless otherwise speciFed we will assume a vector v to be a column vector and so A v makes sense as long as the number of columns of A matches the number of entries in v . Printing matrices on the screen takes up a lot of space, so you may want to use > format compact Enter a matrix into Matlab with the following syntax: > A = [ 1 3 -2 5 ; -1 -1 5 4 ; 0 1 -9 0] Also enter a vector u : > u = [ 1 2 3 4]’ To multiply a matrix times a vector A u use * : > A*u Since A is 3 by 4 and u is 4 by 1 this multiplication is valid and the result is a 3 by 1 vector. Now enter another matrix B using: > B = [3 2 1; 7 6 5; 4 3 2] You can multiply B times A : > B*A but A times B is not deFned and > A*B will result in an error message. You can multiply a matrix by a scalar: > 2*A 28
29 Adding matrices A + A will give the same result: > A + A You can even add a number to a matrix: > A + 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . This should add 3 to every entry of A . Component-wise operations Just as for vectors, adding a ’ . ’ before ‘ * ’, ‘ / ’, or ‘ ^ ’ produces entry-wise multiplication, division and exponentiation. If you enter: > B*B the result will be BB , i.e. matrix multiplication of B times itself. But, if you enter: > B.*B the entries of the resulting matrix will contain the squares of the same entries of B . Similarly if you want B multiplied by itself 3 times then enter: > B^3 but, if you want to cube all the entries of B then enter: > B.^3 Note that B*B and B^3 only make sense if B is square, but B.*B and B.^3 make sense for any size matrix. The identity matrix and the inverse of a matrix The n × n identity matrix is a square matrix with ones on the diagonal and zeros everywhere else. It is called the identity because it plays the same role that 1 plays in multiplication, i.e. AI = A, IA = A, I v = v for any matrix A or vector v where the sizes match. An identity matrix in Matlab is produced by the command: > I = eye(3) A square matrix A can have an inverse which is denoted by A 1 . The deFnition of the inverse is that: AA 1 = I and A 1 A = I.

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## This note was uploaded on 02/09/2012 for the course MATH 344 taught by Professor Young,t during the Fall '08 term at Ohio University- Athens.

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PartII - Part II Linear Algebra c Copyright Todd Young and Martin Mohlenkamp Mathematics Department Ohio University 2007 Lecture 8 Matrices and

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