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Matrices

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A matrix (plural matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. (A matrix is a tensor of rank 2.) The individual items in a matrix are called its elements or entries.

An
m  × n
matrix
A
is commonly written using box brackets (an alternative notation uses large parentheses instead)

An example of a matrix with 3 rows and 5 columns is

Matrix operations

Unary matrix operations

Transpose

The transpose of an
m  × n
matrix
A
is the
n  ×  m
matrix
AT
formed by interchanging the rows and columns of
A
, i.e.

Binary matrix operations

Scalar multiplication

The left scalar multiplication
cA
of a scalar
c
and a matrix
A
is defined as
and the right scalar multiplication
Ac
of a matrix
A
and a scalar
c
is defined as

Matrix addition

The sum
A + B
of two
m  × n
matrices
A
and
B
is calculated entrywise, i.e.

The commutativity of the elements entails the commutativity of matrix addition, i.e.

Matrix multiplication

The matrix multiplication of an
r  × s
matrix
A
and an
s  × t
matrix
B
is defined as
where the product matrix
AB
is an
r  × t
matrix.

Note that matrix multiplication is noncommutative, i.e.

Square matrices

A square matrix is an
n  × n
matrix
where the entries
aii , 1   ≤   i   ≤   n,
constitute the main diagonal of
A
.

The identity matrix

The
n  × n
identity matrix
In
is a square matrix that has 1’s along the main diagonal and 0’s for all other entries. This matrix is often written simply as
I
, and is special in that it acts like 1 in matrix multiplication.

Examples:

     
I1  = 
1
, I2  = 
1   0
0   1
, I3  = 
1   0   0
0   1   0
0   0   1
, I4  = 
1   0   0   0
0   1   0   0
0   0   1   0
0   0   0   1
, I5  = 
1   0   0   0   0
0   1   0   0   0
0   0   1   0   0
0   0   0   1   0
0   0   0   0   1
, ...

Trace

Main article page: Trace

The trace of an
n  × n
square matrix is defined as the sum of the elements on the main diagonal (the diagonal from the upper-left to the lower-right), i.e.
tr (A)  :=
n
i  = 1
  
ai  i  =  a1​1 + a2 2 + + an  n .

Determinant

Main article page: Determinant

The determinant of an
n  × n
square matrix is denoted
| A |
or
det A
. The determinant of a
1 × 1
matrix (a scalar) is defined as (here we avoid the notation
| a |
due to confusion with absolute value)
The determinant of a
2 × 2
matrix is defined as
The determinant of an
n  × n
square matrix is defined as
where the minor
Mij
is defined to be the determinant of the
(n  −  1) × (n  −  1)
matrix obtained from
A
by removing the
i
th row and the
j
th column. The expression
( − 1)i + j Mij
is known as the cofactor of
aij
.

Adjugate

The adjugate of an
n  × n
square matrix
A
is the transpose of the cofactor matrix
C
of
A
, i.e.
adj (A)  :=CT,
where
Cij
is the cofactor of
aij
.
Cij:=  (−1)i  +  j Mij , 1in, 1 ≤   jn,
and where
Mij
is the minor of
aij
.

Inverse

The inverse of an
n  × n
square matrix
A
is an
n  × n
square matrix
A  − 1
defined implicitly as
A A  − 1  =  A  − 1 A  =  In

where

is the
n  × n
identity matrix (the main diagonal entries being 1, all other entries being 0).

A matrix is invertible (regular) if and only if its determinant is nonzero, otherwise the matrix is noninvertible (singular). Laplace’s formula for the inverse matrix is

A  − 1  = 
adj (A)
det (A)
 , det (A) ≠ 0,
where
adj (A)
is the adjugate matrix of
A
.

Eigenvectors and eigenvalues

Main article page: Eigenvectors and eigenvalues

A nonzero scalar
λ
and a nonzero vector
v
satisfying
Av  =  λ v
are called an eigenvalue and an eigenvector of
A
, respectively. The nonzero scalar
λ
is an eigenvalue of an
n  × n
matrix
A
if and only if
A  −  λ In
is not invertible, which is equivalent to
det (A − λ In )  =  0.

See also

External links