# app4 - Poularikas A.D Appendix 4 Matricies and Determinants...

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

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

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

View Full Document
© 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
© 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 .

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 29

app4 - Poularikas A.D Appendix 4 Matricies and Determinants...

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

View Full Document
Ask a homework question - tutors are online