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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
matrix
is commonly written using box brackets (an alternative notation uses large parentheses instead)
![{\displaystyle \mathbf {A} ={\begin{bmatrix}{\begin{array}{cccccc}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\end{array}}\end{bmatrix}}=\left({\begin{matrix}{\begin{array}{cccccc}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\end{array}}\end{matrix}}\right).}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/de83c7c8df418addcd73bd1d917fa99c0c9bd3c7)
An example of a matrix with 3 rows and 5 columns is
![{\displaystyle {\begin{bmatrix}{\begin{array}{rrrrr}-1&0&7&0&7\\34&-3&-67&0&-7\\9&-5&7&1&19\end{array}}\end{bmatrix}}=\left({\begin{matrix}{\begin{array}{rrrrr}-1&0&7&0&7\\34&-3&-67&0&-7\\9&-5&7&1&19\end{array}}\end{matrix}}\right).}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/c549f6bbb206291d2a20ddb538fc84834f59c72e)
Matrix operations
Unary matrix operations
Transpose
The
transpose of an
matrix
is the
matrix
formed by interchanging the
rows and
columns of
, i.e.
![{\displaystyle (\mathbf {A} ^{\rm {T}})_{ij}:=\mathbf {A} _{ji},\quad 1\leq i\leq m,\ 1\leq j\leq n.}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/179c15cd3c295bd187bc01e8e9faae0a3ca8d3e1)
Binary matrix operations
Scalar multiplication
The
left scalar multiplication of a
scalar and a matrix
is defined as
![{\displaystyle (c\mathbf {A} )_{ij}:=c\cdot \mathbf {A} _{ij},\quad 1\leq i\leq m,\ 1\leq j\leq n,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/58b99a207a478a782c82435e9d64c82629afe2c3)
and the
right scalar multiplication of a matrix
and a
scalar is defined as
Matrix addition
The sum
of two
matrices
and
is calculated entrywise, i.e.
![{\displaystyle (\mathbf {A} +\mathbf {B} )_{ij}:=\mathbf {A} _{ij}+\mathbf {B} _{ij},\quad 1\leq i\leq m,\ 1\leq j\leq n.}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/900902f45c0193616598363f743b796a467d6116)
The commutativity of the elements entails the commutativity of matrix addition, i.e.
![{\displaystyle \mathbf {A} _{ij}+\mathbf {B} _{ij}=\mathbf {B} _{ij}+\mathbf {A} _{ij},\quad 1\leq i\leq m,\ 1\leq j\leq n,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/699102779b4c8bcba55ca6cfabe8468c26cd08b0)
![{\displaystyle \implies }](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/913c2e89ea94dfa446f69b056d4bf505e01fcc5f)
![{\displaystyle (\mathbf {A} +\mathbf {B} )_{ij}=(\mathbf {B} +\mathbf {A} )_{ij},\quad 1\leq i\leq m,\ 1\leq j\leq n.}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/618de1a72df4a6156dd325816fe0a133675a6740)
Matrix multiplication
The
matrix multiplication of an
matrix
and an
matrix
is defined as
![{\displaystyle (\mathbf {A} \mathbf {B} )_{ik}:=\sum _{j=1}^{s}\mathbf {A} _{ij}\mathbf {B} _{jk},\quad 1\leq i\leq r,\,1\leq k\leq t,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/1f57c60fffa8e2946a709aad55f01d57c57ba244)
where the product matrix
is an
matrix.
Note that matrix multiplication is noncommutative, i.e.
![{\displaystyle \mathbf {A} \mathbf {B} \neq \mathbf {B} \mathbf {A} .}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/640ffd1d79f3fb1857d09375793fda5ba7b9082d)
Square matrices
A
square matrix is an
matrix
![{\displaystyle {\begin{bmatrix}{\begin{array}{cccccc}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{n1}&a_{n2}&\cdots &a_{nn}\end{array}}\end{bmatrix}},}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/8320a6888d3afed93481ac834357609bc6c57c08)
where the entries
constitute the
main diagonal of
.
The identity matrix
The
identity matrix 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
, and is special in that it acts like
1 in
matrix multiplication.
Examples:
I1 = , I2 = , I3 = , 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
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) := ai i = a11 + a2 2 + ⋯ + an n . |
Determinant
- Main article page: Determinant
The
determinant of an
square matrix is denoted
or
.
The determinant of a
matrix (a
scalar) is defined as (here we avoid the notation
due to confusion with
absolute value)
![{\displaystyle \det {\begin{bmatrix}a_{}\end{bmatrix}}:=a_{}.}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/f9ec541381f1d0acb7f604facff7a824a013eff5)
The determinant of a
matrix is defined as
![{\displaystyle \left|{\begin{matrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{matrix}}\right|:=\det {\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{bmatrix}}:=a_{11}a_{22}-a_{12}a_{21}.}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/ae40996de4292ed5f542fdd7746374fb14de70d7)
The determinant of an
square matrix is defined as
where the
minor is defined to be the determinant of the
matrix obtained from
by removing the
th row and the
th column. The expression
is known as the
cofactor of
.
Adjugate
The
adjugate of an
square matrix
is the
transpose of the
cofactor matrix of
, i.e.
-
where
is the
cofactor of
.
-
C i j := (−1) i + j M i j , 1 ≤ i ≤ n, 1 ≤ j ≤ n, |
and where
is the
minor of
.
Inverse
The
inverse of an
square matrix
is an
square matrix
defined implicitly as
-
where
![{\displaystyle \mathbf {I} _{n}:={\begin{bmatrix}{\begin{array}{cccccc}1&0&\cdots &0\\0&1&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &1\end{array}}\end{bmatrix}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/11bec301df70d783090965c5f6b00b61190e8b92)
is the
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
-
where
is the
adjugate matrix of
.
Eigenvectors and eigenvalues
- Main article page: Eigenvectors and eigenvalues
A nonzero
scalar and a nonzero
vector satisfying
-
are called an
eigenvalue and an
eigenvector of
, respectively. The nonzero scalar
is an eigenvalue of an
matrix
if and only if
is
not invertible, which is equivalent to
-
See also
External links