1_Matrix_Algebra_Review

# 1_Matrix_Algebra_Review - Review of Matrix Algebra Matrices...

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1 Review of Matrix Algebra Matrices •A matrix is a rectangular or square array of values arranged in rows and columns. •An m × n matrix A , has m rows and n columns, and has a general form of 11 12 1 21 22 2 12 × × = …… n n mn mm m n aa a a a Α

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2 Examples of Matrices 358 122 ⎡⎤ = ⎢⎥ ⎣⎦ A 10 36 72 29 = B is a 2 × 3 matrix is a 4 × 2 matrix Vectors •A vector is a matrix with only one column (a column vector ) or only one row (a row vector ).
3 Examples of Vectors 3 2 1 5 ⎡⎤ ⎢⎥ = ⎣⎦ x is a column vector. [ ] 3 215 =− x [ ] 47 5 ′ = y and are row vectors. Scalars • A single number such as 2.4 or 6 is called a scalar . • The elements of a matrix are usually scalars, although a matrix can be expressed as a matrix of smaller matrices.

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4 Vector Addition Example: Given: vector a = vector b = Then: vector c = a + b = a b c = a + b 2 1 1 3 = + + 3 4 1 2 3 1 Multiplication by a Scalar Example: Given: vector a = scalar k = 3 Then: vector c = k a = a c = k a 6 2 13 2 1 = 6 3 2 * 3 1 * 3
5 Transpose • The transpose of a matrix A is the matrix whose columns are rows of A (and therefore whose rows are columns of A ), with order retained, from first to last. • The transpose of A is denoted by A . Example of Matrix Transpose 32 6 71 2 = A 37 21 62 ′ = A Let Then

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6 Matrix Transpose • If A is 2 × 3, then A is 3 × 2. • In general if A is m × n , then A is n × m , and ′ = ij ji aa • The transpose of a row vector is a column vector Partitioned Matrices • The matrix A can be written as a matrix of matrices 11 12 21 22 = AA A • This specification of A is called a partitioning of A , and the matrices A 11 , A 12 , A 21 , A 22 are said to be submatrices of A . A is called a partitioned matrix .
7 Example of Partitioned Matrix 169458 547202 928171 917623 254817 ⎡⎤ ⎢⎥ = ⎣⎦ A Each of the arrays of numbers in the four sections of A engendered by the dashed lines is a matrix. Example of Partitioned Matrix 11 1694 5472 9281 = A 12 58 02 71 = A 21 9176 2548 = A 22 23 17 = A

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8 Trace • The sum of the diagonal elements of a square matrix is called the trace of the matrix, that is, 11 22 1 () tr = =+ + + = " nn n ii i aa a a A Example of Trace 15 9 328 47 6 =− A Let Then tr( A ) = 1 + 2 + 6 = 9
9 Addition and Subtraction • Matrices of the same size are added or subtracted by adding or subtracting corresponding elements. Example of Matrix Addition 248 91 5 ⎡⎤ = ⎢⎥ ⎣⎦ A 15 3 23 7 = B 21 45 83 92( 1 )35( 7 ) 391 1 11 2 2 ++ + += +− + + = AB

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10 Example of Matrix Subtraction 248 91 5 ⎡⎤ = ⎢⎥ ⎣⎦ A 15 3 23 7 = B 21 45 83 92 135( 7 ) 11 5 74 1 2 −− −= = AB Multiplication: Inner Product The inner product
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1_Matrix_Algebra_Review - Review of Matrix Algebra Matrices...

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