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5.2. MATRICES AND DETERMINANTS 167

5.1.5-6. Bounds for complex roots of polynomials with real coefficients.

◦
1 . Routh–Hurwitz criterion. For an algebraic equation (5.1.5.1) with real coefficients, the
number of roots with positive real parts is equal to the number of sign alterations in any of
the two sequences
T 0 , T 1 , T 2 /T 1 , ... , T n /T n–1 ;
T 0 , T 1 , T 1 T 2 , ... , T n–2 T n–1 , a 0 ;
where T m (it is assumed that T m ≠ 0 for all m)are defined by

a n–1
a n–1 a n 0
T 0 = a n > 0, T 1 = a n–1 , a n , T 3 = a n–3 a n–2 a n–1 ,

T 2 =
a n–3 a n–2
a n–5 a n–4 a n–3

a n–1 a n 0 0
a n–1 a n 0 0 0
0
a

a n–3 a n–2 a n–1 n–3 a n–2 a n–1 a n
, T 5 = a n–5 a n–4 a n–3 a n–2 a n–1 , ...
a n

a n–5 a n–4 a n–3
T 4 =
a n–7 a n–6 a n–5 a n–4 a n–3
a n–2
a n–7 a n–6 a n–5 a n–4

a n–9 a n–8 a n–7 a n–6 a n–5
◦
2 . All roots of equation (5.1.5.1) have negative real parts if and only if all T 0 , T 1 , ... , T n
are positive.
3 . All roots of an nth-degree equation (5.1.5.1) have negative real parts if and only if this
◦
is true for the following (n – 1)st-degree equation:
n–1 a n n–2 n–3 a n n–2
a n–1 x + a n–2 – a n–3 x + a n–3 x + a n–4 – a n–5 x + ··· = 0.
a n–1 a n–1
5.2. Matrices and Determinants
5.2.1. Matrices
5.2.1-1. Definition of a matrix. Types of matrices.
A matrix of size (or dimension) m×n is a rectangular table with entries a ij (i = 1, 2, ... , m;
j = 1, 2, ... , n) arranged in m rows and n columns:
⎛ ⎞
a 11 a 12 ··· a 1n
⎜ a 21 a 22 ··· a 2n ⎟
A ≡ ⎜ . . . . ⎟ .
⎝ . . .
. . . . . ⎠
a m1 a m2 ··· a mn
Note that, for each entry a ij , the index i refers to the ith row and the index j to the jth
column. Matrices are briefly denoted by uppercase letters (for instance, A, as here), or by
the symbol [a ij ], sometimes with more details: A ≡ [a ij ](i = 1, 2, ... , m; j = 1, 2, ... , n).
The numbers m and n are called the dimensions of the matrix. A matrix is said to be finite
if it has finitely many rows and columns; otherwise, the matrix is said to be infinite.In what
follows, only finite matrices are considered.

The null or zero matrix is a matrix whose entries are all equal to zero: a ij = 0 (i =
1, 2, ... , m, j = 1, 2, ... , n).
A column vector or column is a matrix of size m ×1.A row vector or row is a matrix
of size 1× n. Both column and row vectors are often simply called vectors.