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2.1 Matrix Operations

2.1 Matrix Operations. Introduction. We can think of a matrix as being like a table of values where all of the entries in a column have something in common, and all of the entries in a row have something in common.

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2.1 Matrix Operations

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  1. 2.1 Matrix Operations

  2. Introduction • We can think of a matrix as being like a table of values where all of the entries in a column have something in common, and all of the entries in a row have something in common. • In chapter 1, the rows were individual linear equations and the columns represented the different variables.

  3. Example • Another Example: Inventory of sweatshirts at a screen-printing shop. • Columns represent sizes of sweatshirts and rows represent colors: (insert colors):

  4. More Background in Matrices • When we use a variable to represent a matrix, we use a capital letter. • Order or Dimension of a matrix: rows x columns (m x n) • 2 x 3 has 3 rows and 2 column:

  5. Row and Column Matrices • Row Matrix: has only 1 row (but can have any number of columns). • Ex. (1 x 6): • Column Matrix: has only 1 column (but can have any number of rows). • Ex (2 x 1):

  6. Matrix Entries • When we want to refer to a particular entry, we refer to it by the row and column in which it falls. (i,j) • Ex. Entry (2,3) is in row 2, column 3 (it is a 3 in matrix D below):

  7. More Background in Matrices • We can also give each entry in a matrix a variable representation in which we indicate its location in the matrix: • We now have another way to give a mtx: A = [aij]

  8. Proofs w/ General matrices • We now have another way to give a mtx: A = [aij] • This will be very helpful when we try to prove something is true for all matrices w/ entries having a specific form. • For instance, if we need A+B, we can say: • A + B = [aij] + [bij] = [aij + bij] (simply discussing the form of each entry rather than writing out entire matrices!!!)

  9. Square Matrix • Square matrix (n x n) has the same number of rows and columns • Main diagonal of a square matrix: Diagonal of entries beginning in the top left and continuing to bottom right.

  10. Equality of Matrices • For 2 matrices to be equal: • the matrices must be of the same size • corresponding entries must be equal

  11. Examples • Which of the following matrices could be equal? • Given that the following matrices are equal, find the values of the variables

  12. Matrix Addition • Matrix Addition is defined so that we can add corresponding items. • Ex. In our screen-printing matrix, we might want to know the combined inventory (of the various categories) from two different stores:

  13. Matrix Addition Example

  14. Commutativity • Is Matrix Addition commutative? • (i.e. does A + B = B + A ?) • Yes, since we are simply adding corresponding entries, each entry will be the same regardless of the order of addition, therefore the matrices will be the same regardless of the order of addition.

  15. Commutativity and Associativity • By the same argument, Matrix Addition is also associative • (i.e. (A+B)+C= A+(B+C)

  16. Zero mtx, negation of mtx • Zero matrix: all entries are 0’s (can just call it 0) • Negative of a matrix, A: all entries in A are multiplied by (-1) • Note that A + (-A) = 0

  17. Example • Ex. Solve the following for X:

  18. Scalar Multiplication • Scalar Multiplication (aX): multiply each entry in X by a. • Example: • Find 2A-3B

  19. Summary of properties • A + B = B + A • A + (B + C) = (A + B) + C • There exists an m x n mtx 0 s.t. 0 + A = A for all A. • For each A there is an m x n mtx (-A) s.t. A + (-A) = 0 • k(A+B) = kA +kB • (k+p)A = kA +pA • (kp)A = k(pA) • 1A = A

  20. Proving the properties • To prove any of the properties, we begin by proving for the individual entries, and then expand to show the implication for the entire matrix.

  21. Proving the properties • Example: property 5: k(A + B) = kA + kB • Take entry aij in A and bij in B • A previous property showed us how to deal with scalar multiplication by entry: • if each entry is like this, we could break into 2 matrices, each multiplied by the scalar k and we get: kA + kB

  22. Example • Solve the following for X and Y:

  23. Transpose of a matrix • The Transpose of an m x n Matrix,A, given by AT: • Is the n x m matrix whose columns are the rows of A (in same order) • Example: • If A = [aij] then AT = [aji]

  24. Some properties of the transpose • If A is an m x n matrix, then AT is an n x m matrix • (AT)T = A • (kA)T = kAT • (A + B)T = AT + BT • (AB)T = BTAT

  25. Proving Property 3 • Property 3: (kA)T = kAT • A = [aij] • kA = [kaij] • (kA)T = [kaji] • k[aji] • = kAT

  26. Proving Property 5 • Property 5: (AB)T = BTAT • A is (m x n), B is (n x p) (i,j) entry of AB =

  27. Proving Property 5 (i,j) entry of BTAT =

  28. Symmetric Matrices • a matrix, A, is Symmetric if AT = A • What do you know about the order of the two matrices? • If A is m x n, then AT is n x m • For A and AT to be equal, they must be of the same order • So m x n = n x m • So they both must be m x m (and therefore square matrices)

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