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Lecture 28: Mathematical Insight and Engineering

Lecture 28: Mathematical Insight and Engineering. Matrices. Matrices 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.

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Lecture 28: Mathematical Insight and Engineering

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  1. Lecture 28: Mathematical Insight and Engineering

  2. Matrices • Matrices 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.

  3. Matrix Operations • Transpose • Multiplication • Exponentiation • Inverse • Determinants • Left division

  4. Transpose • In mathematics texts you will often see the transpose indicated with superscript T • AT • The MATLAB syntax for the transpose is • A'

  5. The transpose switches the rows and columns

  6. Dot Products • The dot product is sometimes called the scalar product • The sum of the results when you multiply two vectors together, element by element.

  7. * * * || || || + + Equivalent statements

  8. Matrix Multiplication • Matrix 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

  9. A: m x n B: n x p

  10. These dimensions must match The resulting matrix will have these dimensions • 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

  11. Matrix Powers • Raising 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

  12. Matrix Inverse • MATLAB offers two approaches • The matrix inverse function • inv(A) • Raising a matrix to the -1 power • A-1

  13. Equivalent approaches to finding the inverse of a matrix A matrix times its inverse is the identity matrix

  14. Not all matrices have an inverse • Called • Singular • Ill-conditioned matrices • Attempting to take the inverse of a singular matrix results in an error statement

  15. Determinants • Related to the matrix inverse • If the determinant is equal to 0, the matrix does not have an inverse • The MATLAB function to find a determinant is • det(A)

  16. A(1,1)*A(2,2)*A(3,3) A(1, 1)*A(2, 2) + A(1, 2)*A(2,3)*A(3,1) - A(1, 2)*A(2, 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) |A| = |A| =

  17. Solutions to Systems of Linear Equations

  18. Using Matrix Nomenclature and AX=B

  19. We can solve this problem using the matrix inverse approach This approach is easy to understand, but its not the more efficient computationally

  20. Matrix left division uses Gaussian elimination, which is much more efficient, and less prone to round-off error

  21. Practice Question Solution: B

  22. Practice Question Solution: B

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