Learn matrix multiplication, scalar multiplication, addition and subtraction

Definition of a matrix

A matrix is a rectangular list of numbers, called elements. Each matrix has rows and columns and its size is called (read "m times n"). Here are a few examples:

Furthermore, matrices are usually noted as integers (A, B, C, etc.). Let the matrix A be a -matrix. Each element and its i.d. (position in the matrix) are usually noted as in the following way:

Addition and subtraction

Summation of matrices can only take place elementally and if the matricies have the same dimensions. Let both and be a pair of -matrices, and that and that then it applies that;

Here we have two examples:

Scalar multiplication

Let be an -matrix. Then the following applies to all vectors and in and each scalar :

Scalar multiplication with a matrix works intuitively. Let be a -matrix summed times. Then it applies that:

and for each element in it applies that:

Matrix multiplication

In order for the multiplication between two matrices to be defined, it is required that the number of columns of the left matrix must correspond to the number of rows of the right matrix. That is, the dimensions of the result matrix are the number of rows of the left matrix times the number of columns of the right matrix. In other words:

But what will be the result matrix of ? We show the simplest multiplication between matrix and vector :

Let's take an example:

We know from the above that the dimensions of the result of become and the result is:

Let's take another example of matrix multiplication where has an additional column and thus is noted as the matrix :

We know from the above that the dimensions of the result of become . The result is:

On a general basis, we conclude that the product of two matrices, and , is calculated by multiplying the rows of with the columns of . So the result is:

where the elements of the matrix become:

Inner and outer products

In linear algebra, we talk about inner and outer products between two vectors of the same dimension, and . These two are defined as follows:

  • inner product: , i.e. a scalar

  • outer product: , a matrix

Take the following example, let:

Then it applies that the inner and outer product are:

Identity, inverse and transpose

The student needs to be aware of the following matrices that we deal with in this section:

  • (the identity matrix)

  • (the inverse of )

  • (the transpose of )

The identity matrix

above refers to the identity matrix, which can be seen as a multidimensional one, an -matrix where all elements are 0 except the diagonal elements, which are all 1.

The identity matrix functions as 1 being the identity operator for all numbers:

namely that:

Note here that multiplication by is commutative.

Inverse

The identity matrix also causes the existence of an inverse matrix, noted as . The following property applies:

If is a multidimensional one, then can be seen as , even if that operation is mathematically illegal.

Transpose

Last but not least, we have the transpose of , which is noted . It can be seen as a rotation of , where its rows become columns, as follows:

The laws of matrix arithmetic

Laws of addition

The following laws apply to matrix addition;

  • (commutative law)

  • (associative law)

Laws of multiplication

Note that the commutative law does not apply to matrix multiplication, i.e.:

However, the following laws do apply:

  • (associative law)

  • (distributive law)

  • (identity)

  • (linearity)

  • (linearity)

Laws for transpose

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