Matrix Algebra - Advanced Engineering Math - Lecture Slides, Slides of Engineering Mathematics

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Matrix algebra

Last week

• The rank of a matrix is
  • The maximal number of linearly independent rows.
  • The maximal number of linearly independent columns.
• The rank of a matrix can be calculated by
  • first reducing the matrix to RREF
  • then counting the number of non-zero rows.

Last week

multiply by B multiply by A

Multiply by

u (^) v w

u w

A is m x n, B is n x p

A B

n x p matrix (^) m x n matrix

m x p matrix

How to define the product of two matrices

The i-th component is

ADDITION AND MULTIPLICATION

OF MATRICES

Matrix equality

• Two matrices are said to be equal if
  1. They have the same number of rows and the same number of columns (i.e. same size).
  2. The corresponding entry are identical.

Matrix multiplication

• Given an mn matrix A and a pq matrix B , their product AB is defined if n=p.
• If n = p, we define their product, say C = AB , by computing the ( i , j )-entry in C as the dot product of the i -th row of A and the j -th row of B. m q mn pq

Examples

is undefined.

but^ is undefined.

Compatibility with

function composition

Multiplied by

Multiplied by

Multiplied by

Order does matter in multiplication

Multiplied by

Rotate 90 degrees Multiplied by

Reflection around x-axis

Multiplied by

Reflection around x-axis Multiplied by

Rotate 90 degrees

not the same

Associativity

• For real numbers, we have (34)5 = 3(45).
  • Multiplication of real numbers is associative.
• For any three matrices A , B , C , it is always true that ( AB ) C = A ( BC ), provided that the multiplications are well-defined. - Multiplication of matrices is associative.

INVERTIBLE MATRIX

Multiplication by identity matrix

• The output and input are identical
• IA = A for any A.
• BI = B for any B.

Multiplied by

is trivial

Invertible matrix

• Given an nn matrix A , if we can find a matrix A ’, such that

then A is said to be invertible, or non-singular.

• The matrix A ’ is called an inverse of A.

Multiplied by A

Multiplied by A ’

Multiplied by I n