Meeting 18 Matrix Operations

Meeting 18 Matrix Operations. Matrix If A is an m x n matrix - that is, a matrix with m rows and n columns – then the scalar entry in the i th row and

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Meeting 18

Matrix Operations

Matrix

If A is an m x n matrix - that is, a matrix with m rows and n columns – then the scalar entry in the i th row and j th column of A is denoted by aij and is called the (i, j)-entry of A.

Matrix is a rectangular array of numbers

Page 3: Meeting 18 Matrix Operations. Matrix If A is an m x n matrix - that is, a matrix with m rows and n columns – then the scalar entry in the i th row and

The diagonal entries in an m x n matrix A =[aij] are � a11, a22, a33, ..., and they form the main diagonal of A.

A diagonal matrix is a square n x n matrix whosenondiagonal entries are zero.

An example is the n x n identity matrix, In.

An m x n matrix whose entries are all zero is a zero matrix and is written as 0.

Page 4: Meeting 18 Matrix Operations. Matrix If A is an m x n matrix - that is, a matrix with m rows and n columns – then the scalar entry in the i th row and

Sums and Scalar MultiplesTwo matrices are equal if they have the same size (i.e., the same number of rows and the same number of columns) and if their corresponding columns are equal, which amounts to saying that their corresponding entries are equal.

If A and B are m x n matrices, then the sum A + B is the m x n matrix whose columns are the sumsof the corresponding columns in A and B.

The sum A + B is defined only when A and B are the same size.

Page 5: Meeting 18 Matrix Operations. Matrix If A is an m x n matrix - that is, a matrix with m rows and n columns – then the scalar entry in the i th row and

Example

Let

Page 6: Meeting 18 Matrix Operations. Matrix If A is an m x n matrix - that is, a matrix with m rows and n columns – then the scalar entry in the i th row and

Scalar Multiple

If r is a scalar and A is a matrix, then the scalar multiple rA is the matrix whose entries are r times the entries in A. As with vectors, -A stands for (-1)A, and A - B is the same as A + (-1)B.

The properties of sums and Scalar Multiplies

Page 7: Meeting 18 Matrix Operations. Matrix If A is an m x n matrix - that is, a matrix with m rows and n columns – then the scalar entry in the i th row and

Example

If A and B are the matrices in the previous example, then

Page 8: Meeting 18 Matrix Operations. Matrix If A is an m x n matrix - that is, a matrix with m rows and n columns – then the scalar entry in the i th row and

Matrix Multiplication

Page 9: Meeting 18 Matrix Operations. Matrix If A is an m x n matrix - that is, a matrix with m rows and n columns – then the scalar entry in the i th row and

Example

Find the product AB where

Solution:

The product AB has size and will take the form

Page 10: Meeting 18 Matrix Operations. Matrix If A is an m x n matrix - that is, a matrix with m rows and n columns – then the scalar entry in the i th row and

To find c11 (the entry in the first row and first column of the product), multiply corresponding entries in the first row of A and the first column of B. That is,

Similarly, to find c12, multiply corresponding entries in the first row of A and the second column of B to obtain

Page 11: Meeting 18 Matrix Operations. Matrix If A is an m x n matrix - that is, a matrix with m rows and n columns – then the scalar entry in the i th row and

Continuing this pattern produces the following results.

The product is

Page 12: Meeting 18 Matrix Operations. Matrix If A is an m x n matrix - that is, a matrix with m rows and n columns – then the scalar entry in the i th row and

Examples

Page 13: Meeting 18 Matrix Operations. Matrix If A is an m x n matrix - that is, a matrix with m rows and n columns – then the scalar entry in the i th row and

Examples

Page 14: Meeting 18 Matrix Operations. Matrix If A is an m x n matrix - that is, a matrix with m rows and n columns – then the scalar entry in the i th row and

Properties of Matrix Multiplication

Page 15: Meeting 18 Matrix Operations. Matrix If A is an m x n matrix - that is, a matrix with m rows and n columns – then the scalar entry in the i th row and

Powers of a MatrixIf A is an n x n matrix and if k is a positive integer, then Ak denotes the product of k copies of A:

Page 16: Meeting 18 Matrix Operations. Matrix If A is an m x n matrix - that is, a matrix with m rows and n columns – then the scalar entry in the i th row and

The Transpose of a MatrixGiven an m x n matrix A, the transpose of A is the n x m matrix, denoted by AT , whose columns are formed from the corresponding rows of A.

Page 17: Meeting 18 Matrix Operations. Matrix If A is an m x n matrix - that is, a matrix with m rows and n columns – then the scalar entry in the i th row and

Exercises

Page 18: Meeting 18 Matrix Operations. Matrix If A is an m x n matrix - that is, a matrix with m rows and n columns – then the scalar entry in the i th row and

Page 19: Meeting 18 Matrix Operations. Matrix If A is an m x n matrix - that is, a matrix with m rows and n columns – then the scalar entry in the i th row and

Page 20: Meeting 18 Matrix Operations. Matrix If A is an m x n matrix - that is, a matrix with m rows and n columns – then the scalar entry in the i th row and

Page 21: Meeting 18 Matrix Operations. Matrix If A is an m x n matrix - that is, a matrix with m rows and n columns – then the scalar entry in the i th row and

Implementation of 2Implementation of 2--D FFT on the Cell ... · Each sub matrix contains 2 rows of real and 2 rows of imaginary dataEach sub matrix contains 2 rows of real and 2

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