Matrices: Matrices After this lecture you should be able to: Appreciate what a matrix is Understand the definitions and terminology of matrices Perform matrix addition and subtraction Perform scalar and matrix multiplication Understand transpose, skew and symmetric matrices Calculate the determinant of a 2x2 matrix Calculate the Inverse of a matrix Solve simultaneous equations using matrices Describe a transformation using matrices Appreciate Eigenvalues and Eigenvectors Learning Objectives
Slide 3: Definition of a matrix A matrix is a set of elements (scalars), organised into ROWS and COLUMNS Rows Columns An mxn matrix has m rows and n columns 2 x 4 3 x 2
Slide 4: Matrix arithmetic Two matrices A =(a ij ) and B =(b ij ) are equal if a ij = b ij for all i an j – (i is defined as rows and j is defined as columns) The zero matrix is defined to be 0 =(0), whose dimensions depend on the context.
Slide 5: Matrix arithmetic Matrix addition and subtraction is as you would expect, however both matrices have to be the same shape and size
Slide 6: Scalar multiplication Multiplication of a matrix by a scalar, is also intuitive Properties of matrix addition and scalar multiplication; If A,B and C are mxn matrices, and c and d are scalars
Slide 7: Matrix multiplication Matrix multiplication is less intuitive and needs a lot more care and concentration If A=[a ij ] mn and B=[b ij ] np then AB=[c ij ] mp where The important thing is that matrices have to “fit” together to be able to be multiplied together. An mxn can only be multiplied by an nxp, resulting in an mxp matrix.
Slide 8: Matrix multiplication The process is best shown by an example: Note in this example BA is not calculable
Slide 9: Matrix multiplication
Slide 10: add elements subtract elements Multiply each row by each column Summary of Basic Operations
Slide 11: Matrix types – transpose, symmetric, skew and identity matrices The transpose of a matrix A = a ij is A T = a ji
Slide 12: Matrix types – transpose, symmetric, skew and identity matrices
Slide 13: Matrix types – transpose, symmetric, skew and identity matrices The matrix is a square matrix and is symmetric The matrix is is skew-symmetric The identity matrix I is an nxn matrix where the leading diagonal (top left to bottom right) are 1 and the remaining entries are 0
Slide 14: The following hold true for any square matrix Matrix types – transpose, symmetric, skew and identity matrices
Slide 16: Determinant of a matrix – leading to the inverse of a matrix The determinant of a square matrix is used to find the inverse of a matrix If ad-bc=0 then the matrix has no inverse and is defined as singular
Slide 17: Determinant of a matrix – leading to the inverse of a matrix
Slide 18: Solving simultaneous equations using matrices Matrices can be used to solve simultaneous equations
Slide 19: TRANSFORMATIONS