Lecture 28: Mathematical Insight and Engineering
description
Transcript of Lecture 28: Mathematical Insight and Engineering
![Page 1: Lecture 28: Mathematical Insight and Engineering](https://reader035.fdocuments.us/reader035/viewer/2022062816/56814ba8550346895db87eb1/html5/thumbnails/1.jpg)
Lecture 28: Mathematical Insight and Engineering
![Page 2: Lecture 28: Mathematical Insight and Engineering](https://reader035.fdocuments.us/reader035/viewer/2022062816/56814ba8550346895db87eb1/html5/thumbnails/2.jpg)
MatricesMatrices are commonly used in engineering computations.A matrix generally has more than one row and more than one column.Scalar multiplication and matrix addition and subtraction are performed element by element.
![Page 3: Lecture 28: Mathematical Insight and Engineering](https://reader035.fdocuments.us/reader035/viewer/2022062816/56814ba8550346895db87eb1/html5/thumbnails/3.jpg)
Matrix OperationsTransposeMultiplicationExponentiationInverseDeterminantsLeft division
![Page 4: Lecture 28: Mathematical Insight and Engineering](https://reader035.fdocuments.us/reader035/viewer/2022062816/56814ba8550346895db87eb1/html5/thumbnails/4.jpg)
TransposeIn mathematics texts you will often see the transpose indicated with superscript T AT
The MATLAB syntax for the transpose is A'
![Page 5: Lecture 28: Mathematical Insight and Engineering](https://reader035.fdocuments.us/reader035/viewer/2022062816/56814ba8550346895db87eb1/html5/thumbnails/5.jpg)
The transpose switches the rows and columns
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
121110987654321
A⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
129631185210741
TA
![Page 6: Lecture 28: Mathematical Insight and Engineering](https://reader035.fdocuments.us/reader035/viewer/2022062816/56814ba8550346895db87eb1/html5/thumbnails/6.jpg)
The dot product is sometimes called the scalar productThe sum of the results when you multiply two vectors together, element by element.
Dot Products
![Page 7: Lecture 28: Mathematical Insight and Engineering](https://reader035.fdocuments.us/reader035/viewer/2022062816/56814ba8550346895db87eb1/html5/thumbnails/7.jpg)
*||
*||
*||
+ +
Equivalent statements
![Page 8: Lecture 28: Mathematical Insight and Engineering](https://reader035.fdocuments.us/reader035/viewer/2022062816/56814ba8550346895db87eb1/html5/thumbnails/8.jpg)
Matrix MultiplicationMatrix multiplication results in an array where each element is a dot product. In general, the results are found by taking the dot product of each row in matrix A with each column in Matrix B
![Page 9: Lecture 28: Mathematical Insight and Engineering](https://reader035.fdocuments.us/reader035/viewer/2022062816/56814ba8550346895db87eb1/html5/thumbnails/9.jpg)
A: m x nB: n x p
€
C(i, j) = A(i,k)*B(k, j)k=1
n
∑
![Page 10: Lecture 28: Mathematical Insight and Engineering](https://reader035.fdocuments.us/reader035/viewer/2022062816/56814ba8550346895db87eb1/html5/thumbnails/10.jpg)
Because matrix multiplication is a series of dot products the number of columns in
matrix A must equal the number of rows in matrix B
For an mxn matrix multiplied by an nxp matrix
m x n n x p
These dimensions must match
The resulting matrix will have these dimensions
![Page 11: Lecture 28: Mathematical Insight and Engineering](https://reader035.fdocuments.us/reader035/viewer/2022062816/56814ba8550346895db87eb1/html5/thumbnails/11.jpg)
Matrix PowersRaising a matrix to a power is equivalent to multiplying itself the requisite number of times A2 is the same as A*A A3 is the same as A*A*A Raising a matrix to a power requires it to have the same number of rows and columns
![Page 12: Lecture 28: Mathematical Insight and Engineering](https://reader035.fdocuments.us/reader035/viewer/2022062816/56814ba8550346895db87eb1/html5/thumbnails/12.jpg)
Matrix InverseMATLAB offers two approaches The matrix inverse function
inv(A) Raising a matrix to the -1 power
A-1
![Page 13: Lecture 28: Mathematical Insight and Engineering](https://reader035.fdocuments.us/reader035/viewer/2022062816/56814ba8550346895db87eb1/html5/thumbnails/13.jpg)
A matrix times its inverse is the identity matrix
Equivalent approaches to finding the inverse of a matrix
![Page 14: Lecture 28: Mathematical Insight and Engineering](https://reader035.fdocuments.us/reader035/viewer/2022062816/56814ba8550346895db87eb1/html5/thumbnails/14.jpg)
Not all matrices have an inverse
Called Singular Ill-conditioned matricesAttempting to take the inverse of a singular matrix results in an error statement
![Page 15: Lecture 28: Mathematical Insight and Engineering](https://reader035.fdocuments.us/reader035/viewer/2022062816/56814ba8550346895db87eb1/html5/thumbnails/15.jpg)
DeterminantsRelated to the matrix inverseIf the determinant is equal to 0, the matrix does not have an inverseThe MATLAB function to find a determinant is det(A)
![Page 16: Lecture 28: Mathematical Insight and Engineering](https://reader035.fdocuments.us/reader035/viewer/2022062816/56814ba8550346895db87eb1/html5/thumbnails/16.jpg)
|A| = A(1, 1)*A(2, 2)- A(1, 2)*A(2, 1)
|A| = A(1,1)*A(2,2)*A(3,3) + A(1, 2)*A(2,3)*A(3,1) + A(1,3)*A(2,1)*a(3,2)- A(3,1)*A(2,2)*A(1,3) - A(3,2)*A(2,3)*A(1,1) - A(3,3)*A(2,1)*A(1,2)
![Page 17: Lecture 28: Mathematical Insight and Engineering](https://reader035.fdocuments.us/reader035/viewer/2022062816/56814ba8550346895db87eb1/html5/thumbnails/17.jpg)
Solutions to Systems of Linear Equations
3 2 103 2 5
1
x y zx y zx y z
+ − =− + + =
− − = −
![Page 18: Lecture 28: Mathematical Insight and Engineering](https://reader035.fdocuments.us/reader035/viewer/2022062816/56814ba8550346895db87eb1/html5/thumbnails/18.jpg)
Using Matrix Nomenclature
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−−
−=
111231123
A⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
zyx
X⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−=
1510
B
and
AX=B
![Page 19: Lecture 28: Mathematical Insight and Engineering](https://reader035.fdocuments.us/reader035/viewer/2022062816/56814ba8550346895db87eb1/html5/thumbnails/19.jpg)
We can solve this problem using the matrix inverse approach
This approach is easy to understand, but its not the more efficient computationally
![Page 20: Lecture 28: Mathematical Insight and Engineering](https://reader035.fdocuments.us/reader035/viewer/2022062816/56814ba8550346895db87eb1/html5/thumbnails/20.jpg)
Matrix left division uses Gaussian elimination, which is much more efficient, and less prone to round-off error
![Page 21: Lecture 28: Mathematical Insight and Engineering](https://reader035.fdocuments.us/reader035/viewer/2022062816/56814ba8550346895db87eb1/html5/thumbnails/21.jpg)
Q. What is the output of the following code fragment? int j; for (j = 0 ; j < 4; j++) { printf(“%d ”, j); j++; } A) 0 1 2 3 B) 0 2 C) 1 3 D) 1 2 3
Practice QuestionSolution: B
![Page 22: Lecture 28: Mathematical Insight and Engineering](https://reader035.fdocuments.us/reader035/viewer/2022062816/56814ba8550346895db87eb1/html5/thumbnails/22.jpg)
Q. What is the output of the following program? #include <stdio.h> int myFunc(int a, int *b); int main( void ) { int a = 4; int b = 5; myFunc(a, &b); printf ("a + b = %d\n", a + b); } int myFunc(int a, int *b) { a = a + 2; *b = *b - 1;
return a; }
A) a + b = 9 B) a + b = 8 C) a + b = 11 D) a + b = 10
Practice QuestionSolution: B