## Matrix And Vector Algebra

C.I Definitions

CJ Matrix Algebra

CJ Trace, Determinant, and Rank

C.4 Matrix Inverses and Solutions to Simultaneous Linear Equations

C.5 Special Types of Square Matrices, Matrix Transformations

C.6 Eigenvectors and Eigenvalues C.7 Functions of Matrices C.8 Vector Calculus C.9 Vectors in Three Dimensions

### C.1 Definitions

A matrix is a rectangular array of scalar entries known as the elements of the matrix. In this book, the scalars are assumed to be real or complex numbers. If all the elements of a matrix are real numbers, the matrix is a real matrix. The matrix

has m rows and n columns, and is referred to as an m X n matrix or as a matrix of order mXn. The equation A =[Atj] should be read as, "A is the matrix whose elements are A^." The first subscript labels the rows of the matrix and the second labels the columns.

Two matrices are equal if and only if they are of the same order and all of the corresponding elements are equal; i.e.,

An n X n matrix is called a square matrix and is usually referred to as being of order n rather than n X n.

The transpose of a matrix is the matrix resulting from interchanging rows and columns. The transpose of A is denoted by AT, and its elements are given by

A = B if and only if Aij=>Bij-,i= l,...,m;j'= 1,...,«

As an example, the transpose of the matrix in Eq. (C-l) is

"2ji

It is clear that the transpose of an mXn matrix is an «Xm matrix, and that the transpose of a square matrix is square. The transpose of the transpose of a matrix is equal to the original matrix:

The adjoint of a matrix, denoted by Ais the matrix whose elements are the complex conjugates of the elements of the transpose of the given matrix,* i.e.,

The adjoint of the adjoint of a matrix is equal to the original matrix:

The adjoint and the transpose of a real matrix are identical.

The main diagonal of a square matrix is the set of elements with row and column indices equal. A diagonal matrix is a square matrix with nonzero elements only on the main diagonal, e.g..

0 0