Introduction Notes

# Introduction Notes - Matrix Theory Math6304 Lecture Notes...

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Matrix Theory, Math6304 Lecture Notes from August 28, 2012 taken by Bernhard Bodmann 0 Course Information Text: R. Horn and C. Johnson, Matrix Analysis, Cambridge University Press, 1885. O ce: PGH 604, 713-743-3851, Mo 2-3pm, We 1-2pm Email: [email protected] Grade: Based on preparation of class notes in LaTeX, rotating note-takers Background knowledge: Linear algebra, calculus Teasers: 1. What if we multiply a matrix on the right by a diagonal matrix? If we have an m × n matrix A with column vectors a 1 , a 2 , . . . , a n , and the diagonal n × n matrix D has diagonal entries d 1 , 1 , d 2 , 2 , . . . , d n,n , then AD = [ a 1 a 2 · · · a n ] D = [ d 1 , 1 a 1 d 2 , 2 a 2 · · · d n,n a n ] . This means the columns of A are multiplied by the corresponding diagonal entries. 2. What if we multiply a matrix on the right by an upper triangular matrix? Take an m × n matrix A with column vectors a 1 , a 2 , . . . , a n and U = ( u i,j ) n i,j =1 with u i,j = 0 if i > j . We see AU = [ u 1 , 1 a 1 u 1 , 2 a 1 + u 2 , 2 a 2 · · · u 1 ,n a 1 + u 2 ,n a 2 + · · · + u n,n a n ] . This means the l th column of A is replaced by a linear combination of the first l columns. 3. What if we multiply a matrix on the left by a diagonal matrix? The rows of the matrix are multiplied by the corresponding diagonal entries. 4. What if we multiply a matrix on the left by an upper triangular matrix? The l th row of the matrix is replaced by a linear combination of rows with indices l . 1

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1 Review 1.1 Range and nullspace 1.1.1 Definition. We write M m,n M m,n ( C ) for the space of all complex m × n matrices. We identify A M m,n and the map A : C n C m , x Ax . Also, we abbreviate M n
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