Matrix Basic Concepts - Temple University
Matrix Basic Concepts
Topics:
? What is a matrix? ? Matrix terminology
? Elements or entries ? Diagonal entries
? Address/location of entries ? Rows and columns ? Size of a matrix ? A column matrix; vectors ? Special types of matrices ? Square ? Diagonal ? Triangular ? Identity matrix ? Operations on matrices ? Equal matrices ? Addition, subtraction ? A number (scalar) times a matrix ? A matrix times a column matrix
Common Matrix Terminology
A matrix is a rectangular array of rows and columns. For example
4 7
2 3
1/ 10
.
We usually name matrices using letters; for instance, letters are often used.)
B
4 7
2 3
1/ 10
.
(Capital
This convention lets us easily refer to a matrix using the letter assigned, here B. The numerical values within a matrix are called elements or entries of the matrix. Each matrix entry has an address/location given by the number of the row and the number of the column in which it appears.
Entries of a matrix are often denoted by a lower case letter of it's assigned name with a pair of subscripts denoting the row, column position of the entry.
Example: For
B
4 7
2 3
1/ 10
we have b11 = 4, b13 = , and b21 = 7.
The entries occupying like numbered row, column position are referred to as diagonal
entries. For the matrix B in the preceding example the diagonal entries are b11 = 4 and b22 = 3.
B
4 7
2 3
1/ 10
The size of a matrix is denoted by listing the number of rows followed by the number of
columns. Matrix B is 2 ? 3; it has two rows and 3 columns. The number of rows is always
stated first.
A column matrix consists of a number of rows and a single column. Each of the following is a column matrix. It is common practice to use lower case letters for column matrices. We have indicated the size of the column below the matrix.
w1
w
w2
,
w3
1
z 2 , 0
9
x
x1 x2
31
41
21
Column matrices are often called vectors. This is from the geometric notions of directed line segments in the plane
4 2
or 3-space. Such geometric vectors are designated by the
coordinates of the end point of the line segment when
we draw it starting from the origin.
Special types of matrices.
1 5 3
?
A square matrix has the same number of rows as column.
3 1
2
8
7
e3
0
12 0.007
2 ? 2
3 ? 3
? A diagonal matrix is a
3 0 0 0 0 0
square matrix in which the non-diagonal entries
7 0
are all zero.
0 9
0 0
0 0 23 0
0 0
0
0
89
0
0
0
0
0 17 0 0 0
Called zero matrices.
? A diagonal matrix with all diagonal entries equal to 1 is called an
1 0 0
1 0
0 1
0 0
1 0
0 etc. 1
identity matrix.
Denoted I2 and I3 respectively.
The n ? n identity is denoted In.
? An upper triangular
? An lower triangular
matrix is a square matrix with all zero entries below the diagonal.
1 0
3
2 4
0 0
2 7 0
0 9 5
matrix is a square matrix with all zero entries above the
diagonal.
6 0 2 1
1 0 0 2 3 0 0 0 0
Matrix Operations
Definition Two m n matrices A = [aij ] and B = [bij ] are said to be equal matrices if aij = bij for 1 i m, 1 j n, that is, if corresponding entries are equal.
Definition If A = [aij ] and B = [bij ] are both m n matrices, then their sum, A + B,
is the matrix whose (i,j)-entry is aij + bij ; that is, we add corresponding entries. The
difference, A - B, is the matrix whose (i,j)-entry is aij - bij ; that is, we subtract
corresponding entries
A
1 6
4 8
,B
10
0
9 1
A
+B
11
6
13 7
and
A
-
B
9
6
5 9
The next operation involves numbers (possibly real or complex), which we call scalars and a matrix. The entries of the matrix could be real numbers, complex numbers, or even functions; we have not restricted the entries of a matrix.
Definition If A = [aij ] is an m n matrix and k is a scalar, then the scalar multiple of A by k, kA, is the m n matrix whose entries are kaij ; that is, each entry of matrix A is multiplied by k.
4 1 i 6 8i 2i 2 12i
7
3 5
1
0
21 35
7
0
,
2i 0 3
2 1
i
0
2 3i 6i
4i 2i
2
4i 6
................
................
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