L5 matrix

Matrix Algebra Definitions Operations

What is it? Matrix algebra is a means of making calculations upon arrays of numbers (or data). Most data sets are matrix-type

Why use it? Matrix algebra makes mathematical expression and computation easier. It allows you to get rid of cumbersome notation, concentrate on the concepts involved and understand where your results come from.

1.1 Matrices Both A and B are examples of matrix. A matrix is a rectangular array of numbers enclosed by a pair of bracket. Why matrix?

1.1 Matrices Matrices can help… Consider the following set of equations: It is easy to show that x = 3 and y = 4. How about solving Matrices can help…

1.1 Matrices In the matrix numbers aij are called elements. First subscript indicates the row; second subscript indicates the column. The matrix consists of mn elements It is called “the m  n matrix A = [aij]” or simply “the matrix A ” if number of rows and columns are understood.

1.1 Matrices Square matrices When m = n, i.e., A is called a “square matrix of order n” or “n-square matrix” elements a11, a22, a33,…, ann called diagonal elements. is called the trace of A.

1.1 Matrices Equal matrices Two matrices A = [aij] and B = [bij] are said to be equal (A = B) iff each element of A is equal to the corresponding element of B, i.e., aij = bij for 1  i  m, 1  j  n. iff pronouns “if and only if” if A = B, it implies aij = bij for 1  i  m, 1  j  n; if aij = bij for 1  i  m, 1  j  n, it implies A = B.

1.1 Matrices Equal matrices Example: and Given that A = B, find a, b, c and d. if A = B, then a = 1, b = 0, c = -4 and d = 2.

1.1 Matrices Zero matrices Every element of a matrix is zero, it is called a zero matrix, i.e.,

Definitions - scalar a scalar is a number (denoted with regular type: 1 or 22)

Vectors and Matrices Matrix is an array of numbers with dimensions M (rows) by N (columns) 3 by 6 matrix element 2,3 is (3) Vector can be considered a 1 x M matrix

Definitions - vector Vector: a single row or column of numbers denoted with bold small letters row vector a = column vector b =

Definitions - Matrix A matrix is an array of numbers A = Denoted with a bold Capital letter All matrices have an order (or dimension): that is, the number of rows  the number of columns. So, A is 2 by 3 or (2  3).

What is a matrix? A Matrix is just rectangular arrays of items A typical matrix is a rectangular array of numbers arranged in rows and columns.

Sizing a matrix By convention matrices are “sized” using the number of rows (m) by number of columns (n).

“Special” Matrices Square matrix: a square matrix is an mxn matrix in which m = n. Vector: a vector is an mxn matrix where either m OR n = 1 (but not both).

“Special” Matrices Scalar: a scalar is an mxn matrix where BOTH m and n = 1. Zero matrix: an mxn matrix of zeros. Identity Matrix: a square (mxm) matrix with 1s on the diagonal and zeros everywhere else.

Types of Matrix Identity matrices - I Diagonal Symmetric Diagonal matrices are (of course) symmetric Identity matrices are (of course) diagonal

Identity matrix A square matrix whose elements aij = 0, for i > j is called upper triangular, i.e., A square matrix whose elements aij = 0, for i < j is called lower triangular, i.e.,

1.3 Types of matrices Identity matrix Both upper and lower triangular, i.e., aij = 0, for i  j , i.e., is called a diagonal matrix, simply

1.3 Types of matrices Identity matrix In particular, a11 = a22 = … = ann = 1, the matrix is called identity matrix. Properties: AI = IA = A Examples of identity matrices: and

1.3 Types of matrices Symmetric matrix A matrix A such that AT = A is called symmetric, i.e., aji = aij for all i and j. A + AT must be symmetric. Why? Example: is symmetric. A matrix A such that AT = -A is called skew-symmetric, i.e., aji = -aij for all i and j. A - AT must be skew-symmetric. Why?

We’ll see that orthogonal matrix represents a rotation in fact! 1.3 Types of matrices Orthogonal matrix A matrix A is called orthogonal if AAT = ATA = I, i.e., AT = A-1 Example: prove that is orthogonal. Since, . Hence, AAT = ATA = I. Can you show the details? We’ll see that orthogonal matrix represents a rotation in fact!

1.4 Properties of matrix (AB)-1 = B-1A-1 (AT)T = A and (lA)T = l AT (A + B)T = AT + BT (AB)T = BT AT

Definitions A square matrix is a matrix that has the same number of rows and columns (n  n)

Matrix Equality Two matrices are equal if and only if they both have the same number of rows and the same number of columns their corresponding elements are equal

Matrix Rank Matrix Rank: the rank of a matrix is the maximum number of linearly independent vectors (either row or column) in a matrix Full Rank: A matrix is considered full rank when all vectors are linearly independent

Transposing a Matrix Matrix Transpose: is the mxn matrix obtained by interchanging the rows and columns of a matrix (converting it to an nxm matrix)