What is a Matrix? A matrix definition An array of numbers a1n aml amn with m rows and n columns is an m x n matrix. Element aij is located in position (i,j). The element aij are scalars,namely,real or complex numbers. The set of real numbers is R;the set of complex numbers is C. Remarks: 1.A matrix is a rectangular pattern of numbers-we usually enclose the numbers with brackets. 2.These elements can be any numbers we choose-positive, negative,zero,fractions,decimals,and so on. 3.To refer briefly to a specific matrix we might label it,usually with a capital letter. 命电有这女子 Matrix Theory Matrices -4/13
What is a Matrix? A matrix definition An array of numbers A = ⎛ ⎜ ⎝ a11 . . . a1n ⋮ ⋮ am1 . . . amn ⎞ ⎟ ⎠ with m rows and n columns is an m × n matrix. ▸ Element aij is located in position (i, j). ▸ The element aij are scalars, namely, real or complex numbers. ▸ The set of real numbers is R; the set of complex numbers is C. Remarks: 1. A matrix is a rectangular pattern of numbers –we usually enclose the numbers with brackets. 2. These elements can be any numbers we choose - positive, negative, zero, fractions, decimals, and so on. 3. To refer briefly to a specific matrix we might label it, usually with a capital letter. Matrix Theory Matrices - 4/13
What is a Matrix? An m x n matrix can encode information about m objects (each described by n features); the behavior of a discretized differential operator on a finite element mesh 命电有这女子 Matrix Theory Matrices -5/13
What is a Matrix? An m × n matrix can encode information about ▸ m objects (each described by n features); ▸ the behavior of a discretized differential operator on a finite element mesh An n × n positive-definite matrix can encode ▸ the correlations between all pairs of n objects; ▸ the edge-connectivity between all pairs of nodes in a social network; ▸ and so on. Matrix Theory Matrices - 5/13
What is a Matrix? An m x n matrix can encode information about m objects (each described by n features); the behavior of a discretized differential operator on a finite element mesh An n x n positive-definite matrix can encode the correlations between all pairs of n objects; the edge-connectivity between all pairs of nodes in a social network; and so on. 奇电有这头 Matrix Theory Matrices -5/13
What is a Matrix? An m × n matrix can encode information about ▸ m objects (each described by n features); ▸ the behavior of a discretized differential operator on a finite element mesh An n × n positive-definite matrix can encode ▸ the correlations between all pairs of n objects; ▸ the edge-connectivity between all pairs of nodes in a social network; ▸ and so on. Matrix Theory Matrices - 5/13
What is a Matrix? We write 奇电有这头 Matrix Theory Matrices -6/13
What is a Matrix? We write ▸ A ∈ R m×n : if A is an m × n matrix whose elements are real numbers. ▸ A ∈ C m×n : if A is an m × n matrix whose elements are complex numbers. Of course, R m×n ⊂ C m×n . Refer to the size of a matrix, we state its number of rows and number of columns, in that order. –That is, Matrix A has two rows and two columns; we write that it is a 2 × 2 matrix and say that it is a ‘two by two’ matrix. For instance, A = ( 1 2 3 4 5 6 7 8 ) is a 2 × 4 matrix with elements a13 = 3 and a24 = 8. Matrix Theory Matrices - 6/13
What is a Matrix? We write AeRmxn:if A is an mx n matrix whose elements are real numbers. 命电有这女子 Matrix Theory Matrices -6/13
What is a Matrix? We write ▸ A ∈ R m×n : if A is an m × n matrix whose elements are real numbers. ▸ A ∈ C m×n : if A is an m × n matrix whose elements are complex numbers. Of course, R m×n ⊂ C m×n . Refer to the size of a matrix, we state its number of rows and number of columns, in that order. –That is, Matrix A has two rows and two columns; we write that it is a 2 × 2 matrix and say that it is a ‘two by two’ matrix. For instance, A = ( 1 2 3 4 5 6 7 8 ) is a 2 × 4 matrix with elements a13 = 3 and a24 = 8. Matrix Theory Matrices - 6/13