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Chapter 1.2 MatrixAlgebra

# Chapter 1.2 MatrixAlgebra - Outline Matrices Matrix Algebra...

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Outline Matrices Matrix Algebra Matrix Equations Trace and Transpose Matrix Algebra Math 2270 K. J. Platt, Ph.D. Department of Mathematics Snow College Spring 2014

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Outline Matrices Matrix Algebra Matrix Equations Trace and Transpose Outline 1 Matrices 2 Matrix Algebra Matrix Addition, Subtraction, and Scalar Multiplication Matrix Multiplication 3 Matrix Equations 4 Trace and Transpose The Trace of a Matrix The Transpose of a Matrix
Outline Matrices Matrix Algebra Matrix Equations Trace and Transpose Matrix Notation and Terminology Matrices first came up in the context of systems of equations. Matrices are useful in their own right, outside of the context of linear systems. We proceed to abstract the concept and analyze its properties. Definition A matrix is a rectangular array A of numbers. If A has m rows and n columns, then we say that the matrix has size m × n . The number in row i and column j is denoted a ij , and we will write A = [ a ij ] or A = [ a ij ] m × n . Definition Two matrices A = [ a ij ] and B = [ b ij ] are equal if and only if they are the same size and a ij = b ij for each i and j .

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Outline Matrices Matrix Algebra Matrix Equations Trace and Transpose Special Matrices A matrix with n rows and n columns is called a square matrix . A matrix with only one row is a row vector . A matrix with only one column is called a column vector . Row and column vectors are usually denoted by a lowercase boldface letter. The entries a 11 , a 22 , . . . , a nn in a square matrix are called the diagonal entries . Sometimes this is called the main diagonal to distinguish it from the other diagonal in the matrix. A matrix whose only nonzero entries lie on the diagonal is called a diagonal matrix . A matrix consisting only of zero entries is called a zero matrix . A scalar is a real number.
Outline Matrices Matrix Algebra Matrix Equations Trace and Transpose Examples Discuss the following matrices. A = 1 2 3 4 - 1 - 3 - 4 - 2 1 - 1 0 1 0 2 3 4 B = 0 0 0 0 0 0 a = 2 - 3 0 1 C = 1 2 3 0 1 2 0 0 0 D = 1 0 0 0 0 0 0 0 0 0 5 0 0 0 0 - 2 b = 5 - 4 - 3 0 9

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Outline Matrices Matrix Algebra Matrix Equations Trace and Transpose Matrix Operations Definition Let A = [ a ij ] and B = [ b ij ] be two m × n matrices, and let λ be a scalar.
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