456

MATRICES AND DETERMINANTS

[APP. B

Unit matrix. A diagonal matrix having every diagonal element unity. Null matrix. A matrix in which every element is zero. Square matrix. A matrix in which the number of rows is equal to the number of columns; dðAÞ ¼ n  n. Symmetric matrix. Given 2 3 a11 a12 a13 . . . a1n 6 a21 a22 a23 . . . a2n 7 7 dðAÞ ¼ m  n A6 4 ... ... ... ... ... 5 am1 am2 am3 . . . amn the transpose of A is 2

a11 6 a12 6 AT  6 6 a13 4 ... a1n

a21 a22 a23 ... a2n

a31 a32 a33 ... a3n

... ... ... ... ...

Thus, the rows of A are the columns of AT , and symmetric matrix must then be square. Hermitian matrix. Given 2 a11 a12 6 a21 a22 A6 4 ... ... am1 am2

3 am1 am2 7 7 am3 7 7 ... 5 amn vice versa.

dðAT Þ ¼ n  m

Matrix A is symmetric if A ¼ AT ; a

3 a1n a2n 7 7 ... 5 amn

a13 a23 ... am3

... ... ... ...

a13 a23 ... am3

3 . . . a1n . . . a2n 7 7 ... ... 5 . . . amn

the conjugate of A is 2

a11 6 a21 A  6 4 ... am1

a12 a22 ... am2

Matrix A is hermitian if A ¼ ðA ÞT ; that is, a hermitian matrix is a square matrix with real elements on the main diagonal and complex conjugate elements occupying positions that are mirror images in the main diagonal. Note that ðA ÞT ¼ ðAT Þ . Nonsingular matrix. An n  n square matrix A is nonsingular (or invertible) if there exists an n  n square matrix B such that AB ¼ BA ¼ I where I is the n  n unit matrix. The matrix B is called the inverse of the nonsingular matrix A, and we write B ¼ A1 . If A is nonsingular, the matrix equation Y ¼ AX of Section B1 has, for any Y, the unique solution X ¼ A1 Y

B3

MATRIX ARITHMETIC

Addition and Subtraction of Matrices Two matrices of the same order are conformable for addition or subtraction; two matrices of diﬀerent orders cannot be added or subtracted. The sum (diﬀerence) of two m  n matrices, A ¼ ½aij  and B ¼ ½bij , is the m  n matrix C of which each element is the sum (diﬀerence) of the corresponding elements of A and B. Thus, A  B ¼ ½aij  bij .

Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An
Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An