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.

Matrix Operations Transpose Multiplication Exponentiation Inverse Determinants Left division

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

The transpose switches the rows and columns

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

* || * * + + Equivalent statements

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

A: m x n B: n x p

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

Matrix Powers Raising a matrix to a power is equivalent to multiplying itself the requisite number of times A 2 is the same as A*A A 3 is the same as A*A*A Raising a matrix to a power requires it to have the same number of rows and columns

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

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

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

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)

|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)

Solutions to Systems of Linear Equations

Using Matrix Nomenclature and AX=B

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

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

Practice Question Solution: B

Practice Question Solution: B