Matrix Addition and Scaling
Addition of Matrices
Given two linear transformations , we define their sum as follows: For any vector in ,
The definition makes sense because for any vector in , both and are vectors in and hence we can add them together.
It is easy to see that is again a linear transformation from to .
In the previous section, we already learned that any linear transformation can be uniquely represented by an m x n matrix. What can we say about the relationship among the matrices for and ?
First of all, we write down the matrices for and respectively. Let be the standard basis for . Then are the n columns of the matrix for . Similarly, are the n columns of the matrix for . The matrices can be expressed as follows:
They are m x n matrices because each column is a column vector in . Now, we consider the matrix for S+T. It is formed by column vectors . By definition, is simply for . Therefore, the matrix for is
Therefore, it is natural to define to be the above m x n matrix.
More explicitly, we can also write down an m x n matrix entry by entry as follows:
Usually, we can refer to a specific entry in an m x n matrix by the indices in the subscript - is the entry of the matrix on the row and column. According to the above definition can be obtained by adding the corresponding column vectors of and together, which implies that each entry in is just the sum of the corresponding entries of and i.e. we have
This is how we compute the addition of two matrices of the same size. For matrices with different sizes, it does not make sense to define the addition of them.
Scaling a Matrix
Given a linear transformation and a real number , we define the scaling of by as follows: For any vector in ,
Since is a vector in , it makes sense to scale it by . Morever, it is easy to see that is also a linear transformation from to . The matrix for is as follows:
By definition, for . Therefore, each entry of this matrix is simply k times the corresponding entry of the matrix for . Let A be the matrix for S, then it is natural to define kA as follows:
This definition is consistent with the scaling of column vectors if we regard column vectors as n x 1 matrices.
Basic Properties of Matrix Operations
Two matrices are said to be equal if they have the same size and their corresponding entries are equal.
From the definitions above, it is not difficult to see that the usual rules algebra apply to sums and scaling of matrices: Let and be matrices of the same size, and let and be real numbers.
- (Note: Here "0" means zero matrix i.e. a matrix with all entries zero)
Let and . Find .
Suppose and are two 2 x 2 matrices such that they satisfy the following matrix equations: Find and .