Introduction Notes - Matrix Theory Math6304 Lecture Notes...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern