Unizor - Matrices - Basic Operations

Let's describe the basic operations with matrices.

Addition
This is a rather formal operation. It's easy to define, but it's not directly related to matrices viewed from the standpoint of linear transformation.
Two matrices of the same size (that is, the same number of rows and columns) can be added, each element of one matrix added to a corresponding element with the same row number and column number of another matrix getting a new matrix - a sum of two original ones.
Here is how it looks for two 3x3 matrices:
a11 a12 a13
a21 a22 a23
a31 a32 a33
+
b11 b12 b13
b21 b22 b23
b31 b32 b33
=
a11+b11 a12+b12 a13+b13
a21+b21 a22+b22 a23+b23
a31+b31 a32+b32 a33+b33
In general, if one MxN matrix [aij] (row index i going from 1 to M, column index j going from 1 to N) is added to another matrix [bij] of the same size, the result will be an MxN matrix with elements [cij] where each cij = aij+bij.
Obviously, matrix addition is a commutative and associative operation since it is reduced to an operation of addition of numbers.

Multiplication by a constant
(Scalar multiplication)
This is a rather formal operation. It's easy to define, but it's not directly related to matrices viewed from the standpoint of linear transformation.
Any matrix can be multiplied by a constant producing a matrix of the same size with each element being a product of the original matrix element and that constant.
Here is how it looks for a 3x3 matrix A:
a11 a12 a13
a21 a22 a23
a31 a32 a33
times K =
K·a11 K·a12 K·a13
K·a21 K·a22 K·a23
K·a31 K·a32 K·a33
Obviously, since this operation on matrices is reduced to an operation on individual elements, scalar multiplication of a matrix is commutative (in a sense that a matrix multiplied by a constant equals to a constant multiplied by a matrix)
K·A = A·K
and associative (in terms that consecutive multiplication of a matrix by two constants equals to a product of a matrix multiplied by one constant, the product of two original constants)
K·(L·A) = (K·L)·A
For the same reason, scalar multiplication of matrices is distributive relative to matrix addition and scalar addition:
K·(A+B) = K·A + K·B
(K+L)·A = K·A + L·A
where K and L are scalars, A and B are matrices of the same size.

Transposition
This is an operation of exchanging rows and columns. More precisely, if MxN matrix A=[Aij] (row index i is from 1 to M, column index j is from 1 to N) is transposed into NxM matrix B=[Bkl] (row index k is from 1 to N, column l is from 1 to M) then
Bkl = Alk.
The corresponding notation for the operation of transposition is
B = A^T
For example, if 3x2 matrix A looks like this
a11 a12
a21 a22
a31 a32
then 2x3 matrix B=A^T looks like this
a11 a21 a31
a12 a22 a32
Obviously, applied twice, transposition would result in the original matrix:
(A^T)^T = A
Other trivial properties of the operation of transposition are:
(A+B)^T = A^T + B^T
(K·A)^T = K·(A^T)

[ZorShekhtman]

Let's describe the basic operations with matrices.

Addition
This is a rather formal operation. It's easy to define, but it's not directly related to matrices viewed from the standpoint of linear transformation.
Two matrices of the same size (that is, the same number of rows and columns) can be added, each element of one matrix added to a corresponding element with the same row number and column number of another matrix getting a new matrix - a sum of two original ones.
Here is how it looks for two 3x3 matrices:
a11 a12 a13
a21 a22 a23
a31 a32 a33
+
b11 b12 b13
b21 b22 b23
b31 b32 b33
=
a11+b11 a12+b12 a13+b13
a21+b21 a22+b22 a23+b23
a31+b31 a32+b32 a33+b33
In general, if one MxN matrix [aij] (row index i going from 1 to M, column index j going from 1 to N) is added to another matrix [bij] of the same size, the result will be an MxN matrix with elements [cij] where each cij = aij+bij.
Obviously, matrix addition is a commutative and associative operation since it is reduced to an operation of addition of numbers.

Multiplication by a constant
(Scalar multiplication)
This is a rather formal operation. It's easy to define, but it's not directly related to matrices viewed from the standpoint of linear transformation.
Any matrix can be multiplied by a constant producing a matrix of the same size with each element being a product of the original matrix element and that constant.
Here is how it looks for a 3x3 matrix A:
a11 a12 a13
a21 a22 a23
a31 a32 a33
times K =
K·a11 K·a12 K·a13
K·a21 K·a22 K·a23
K·a31 K·a32 K·a33
Obviously, since this operation on matrices is reduced to an operation on individual elements, scalar multiplication of a matrix is commutative (in a sense that a matrix multiplied by a constant equals to a constant multiplied by a matrix)
K·A = A·K
and associative (in terms that consecutive multiplication of a matrix by two constants equals to a product of a matrix multiplied by one constant, the product of two original constants)
K·(L·A) = (K·L)·A
For the same reason, scalar multiplication of matrices is distributive relative to matrix addition and scalar addition:
K·(A+B) = K·A + K·B
(K+L)·A = K·A + L·A
where K and L are scalars, A and B are matrices of the same size.

Transposition
This is an operation of exchanging rows and columns. More precisely, if MxN matrix A=[Aij] (row index i is from 1 to M, column index j is from 1 to N) is transposed into NxM matrix B=[Bkl] (row index k is from 1 to N, column l is from 1 to M) then
Bkl = Alk.
The corresponding notation for the operation of transposition is
B = A^T
For example, if 3x2 matrix A looks like this
a11 a12
a21 a22
a31 a32
then 2x3 matrix B=A^T looks like this
a11 a21 a31
a12 a22 a32
Obviously, applied twice, transposition would result in the original matrix:
(A^T)^T = A
Other trivial properties of the operation of transposition are:
(A+B)^T = A^T + B^T
(K·A)^T = K·(A^T)

[ZorShekhtman]

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