MATRICES AND DETERMINANTS
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 A6 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 A6 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
... ... ... ...
a13 a23 ... am3
3 . . . a1n . . . a2n 7 7 ... ... 5 . . . amn
the conjugate of A is 2
a11 6 a21 A 6 4 ... am1
a12 a22 ... am2
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 A1 . If A is nonsingular, the matrix equation Y ¼ AX of Section B1 has, for any Y, the unique solution X ¼ A1 Y
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 .