app4 - Poularikas, A.D. Appendix 4: Matricies and...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
Poularikas, A.D . Appendix 4 : Matricies and Determinants .” The Transforms and Applications Handbook: Second Edition. Ed. Alexander D. Poularikas Boca Raton: CRC Press LLC, 2000
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
© 2000 by CRC Press LLC Appendix 4: Matrices and Determinants 1 General Defnitions 1.1. A matrix is an array of numbers consisting of m rows and n columns. It is usually denoted by a boldface capital letter, e.g., A Σ M . 1.2. The ( i , j ) element of a matrix is the element occurring in row i and column j . It is usually denoted by a lowercase letter with subscripts, e.g., a ij σ m . Exceptions to this convention will be stated where required. 1.3. A matrix is called rectangular if m (number of rows) n (number of columns). 1.4. A matrix is called square if m = n . 1.5a. In the transpose of a matrix A, denoted by A , the element in the j th row and i th column of A is equal to the element in the i th row and j th column of A . Formally, ( A ) = ( A ) ji where the symbol ( A ) denotes the ( i j )th element of A . 1.5b. The Hermitian conjugate of a matrix A, denoted by A H or A , is obtained by transposing A and replacing each element by its conjugate complex. Hence, if a kl = u + i υ , then ( A H ) u i , where typical elements have been denoted by ( k l ) to avoid confusion with i =. 1.6a. A square matrix is called symmetric if A = A . 1.6b. A square matrix is called Hermitian if A = A H . 1.7. A matrix with m rows and one column is called a column vector and is usually denoted by boldface, lowercase letters, e.g., β x a . 1.8. A matrix with one row and n columns is called a row vector and is usually denoted by a primed, boldface, lowercase letter, e.g., 1
Background image of page 2
© 2000 by CRC Press LLC a c µ . 1.9. A matrix with one row and one column is called a scalar and is usually denoted by a lowercase letter, occasionally italicized. 1.10. The diagonal extending from upper left (NW) to lower right (SE) is called the principal diagonal of a square matrix. 1.11a. A matrix with all elements above the principal diagonal equal to zero is called a lower triangular matrix. Example 1.11b. The transpose of a lower triangular matrix is called an upper triangular matrix. 1.12. A square matrix with all off-diagonal elements equal to zero is called a diagonal matrix, denoted by the letter D with a subscript indicating the typical element in the principal diagonal. Example 2 Addition, Subtraction, and Multiplication 2.1. Two matrices A and B can be added (subtracted) if the number of rows (columns) in A equals the number of rows (columns) in B. A ± B = C implies 2.2. Multiplication of a matrix or vector by a scalar implies multiplication of each element by the scalar. If B γ A , then b ij a for all elements. 2.3a. Two matrices A and B can be multiplied if the number of columns in A equals the number of rows in B. T = t tt ttt 11 21 22 31 32 33 00 0 is lower triangular .
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 29

app4 - Poularikas, A.D. Appendix 4: Matricies and...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online