meeting 18 matrix operations. matrix if a is an m x n matrix - that is, a matrix with m rows and n...
TRANSCRIPT
![Page 1: 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](https://reader036.vdocuments.us/reader036/viewer/2022062518/5697bf8b1a28abf838c8b155/html5/thumbnails/1.jpg)
Meeting 18
Matrix Operations
![Page 2: 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](https://reader036.vdocuments.us/reader036/viewer/2022062518/5697bf8b1a28abf838c8b155/html5/thumbnails/2.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022062518/5697bf8b1a28abf838c8b155/html5/thumbnails/3.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022062518/5697bf8b1a28abf838c8b155/html5/thumbnails/4.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022062518/5697bf8b1a28abf838c8b155/html5/thumbnails/5.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022062518/5697bf8b1a28abf838c8b155/html5/thumbnails/6.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022062518/5697bf8b1a28abf838c8b155/html5/thumbnails/7.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022062518/5697bf8b1a28abf838c8b155/html5/thumbnails/8.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022062518/5697bf8b1a28abf838c8b155/html5/thumbnails/9.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022062518/5697bf8b1a28abf838c8b155/html5/thumbnails/10.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022062518/5697bf8b1a28abf838c8b155/html5/thumbnails/11.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022062518/5697bf8b1a28abf838c8b155/html5/thumbnails/12.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022062518/5697bf8b1a28abf838c8b155/html5/thumbnails/13.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022062518/5697bf8b1a28abf838c8b155/html5/thumbnails/14.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022062518/5697bf8b1a28abf838c8b155/html5/thumbnails/15.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022062518/5697bf8b1a28abf838c8b155/html5/thumbnails/16.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022062518/5697bf8b1a28abf838c8b155/html5/thumbnails/17.jpg)
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](https://reader036.vdocuments.us/reader036/viewer/2022062518/5697bf8b1a28abf838c8b155/html5/thumbnails/18.jpg)
![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](https://reader036.vdocuments.us/reader036/viewer/2022062518/5697bf8b1a28abf838c8b155/html5/thumbnails/19.jpg)
![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](https://reader036.vdocuments.us/reader036/viewer/2022062518/5697bf8b1a28abf838c8b155/html5/thumbnails/20.jpg)
![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](https://reader036.vdocuments.us/reader036/viewer/2022062518/5697bf8b1a28abf838c8b155/html5/thumbnails/21.jpg)