University Mathematics Handbook (2015)
X. Algebra
Chapter 5. Matrices
5.1 Definition and Types of Matrices
a. A table of 
real (complex) numbers arranged in 
rows and 
columns
is an order real (complex) matrix.
b. 
is the set of all matrices with 
field elements.
c. 
is the 
-row vector on 
, 
.
d. 
is the 
-column vector in , 
.
e. The transposed matrix of , 
, results from switching the rows to columns. That is, if 
then 
and
f. A matrix where the number of rows equals the number of columns is called -th order square matrix.
g. A matrix in which all of its elements are zeroes is a zero matrix.
h. A square matrix 
is called diagonal matrix if for every 
, that is, all elements except those on the main diagonal are zeroes. It denoted 
.
i. A diagonal matrix is scalar if all terms of the main diagonal equal each other.
j. An identity matrix is a scalar matrix where the elements of the diagonal equal 
. An -th order identity matrix is denoted 
.
k. Upper (lower) triangular matrix is a square matrix where all elements below (above) the main diagonal are zeroes. In other words, matrix 
is upper triangular if 
for every 
, and lower triangular when 
for every 
.
l. Square matrix 
is called symmetric if 
for every 
, that is, 
.
m. If, in matrix 
, 
, for every 
that is, 
then, it is called anti-symmetric matrix. In an anti-symmetric matrix, element 
situated on the main diagonal equal zero.
n. Any square matrix can be presented as the sum of a symmetric and an anti-symmetric matrix. That is,
, when 
is a symmetric matrix, and 
, an anti-symmetric one.
o. Complex matrix 
is a conjugate transpose of matrix . It means that 
is obtained by taking conjugate complex of every element of .
p. Matrix is called Hermitian if 
.
q. Matrix is called anti-Hermitian if 
.
r. Matrix is called normal if 
.
s. Matrix is called unitary if 
.
t. A real unitary matrix is called an orthogonal matrix.
5.2 Algebraic Operations on Matrices
a. Two matrices are equal if they are of the same order and the elements of the same locations are equal.
b. The product of matrix multiplied by scalar 
is matrix 
resulted from the multiplication of all elements by 
.
c. The sum of matrices and 
of is the matrix resulted from summing their corresponding elements. That is: 
.
d. The inner product of vectors 
and 
of 
is scalar
.
e. The multiplication 
of matrices and 
is 
order matrix 
, when its 
element is the inner multiplication of i-row vector of matrix by -column vector of matrix :
Example: 
f. If 
, then:
1. 
2. 
3. 
4. 
5. 
g. The trace of square matrix 
is the sum of elements on its main diagonal 
.
1. 
2. 
5.3 Row Space and Columns Space
a. The row space of matrix is the span of m rows on .
b. The column space of 
is the span of its columns.
c. The row space dimension of is equal to its column space dimension and is called the rank of matrix A, 
.
5.4 Elementary Row Operations, Staircase Matrix
a. The elementary row operations of matrix 
are:
1. Interchanging row with row 
: 
.
2. Multiplying row by a non-zero scalar, 
, 
.
3. Adding 
times row to row : 
.
b. Matrix is a staircase matrix if:
1. All rows of zeroes, if it has such, are at the bottom.
2. Every non-zero first element is right of the non-zero first element of the last row.
c. Number of non-zero rows in the echelon form matrix resulted from matrix by elementary row operation equals to the rank of matrix .
5.5 Invertible Matrix
a. Square matrix is invertible if there exists a matrix 
such that
where 
is identity matrix.
Matrix is called inverse of and is denoted 
.
b. If 
are invertible matrices, then:
1. in invertible and 
2. Multiplication is invertible, and 
.
c. To find , we construct matrix 
, and using elementary row operations, get from matrix to identity matrix , getting, instead of .
Example: Find the to matrix 
The result is 
5.6 One-sided Invertibility
a. Matrix of order of is left invertible of there exists matrix of 
order, such that 
. Matrix is called the left inverse of .
b. Matrix is right invertible if there exists matrix 
, such that 
. is called the right inverse of .