I hold both a bachelor's and a master's degree in applied mathematics.
What Is a Matrix?
A matrix is a rectangular array of numbers. It can be used to do linear operations such as rotations, or it can represent systems of linear inequalities.
A matrix is generally denoted with the letter A, which has n rows and m columns. Therefore, a matrix has n*m entries. We also speak of an n times m matrix, or in short, an n x m matrix.
Any linear system can be written down with the use of a matrix. Let's look at the following system:
a + b + c = 3
2a + 3c = 4
5b - 2c = 7
This can be written down as a matrix times a vector equals a vector. This is shown in the picture below.
This gives a much clearer view of the system. In this case, the system consists of only three equations. Therefore, the difference is not so big. However, the matrix notation becomes preferred when the system has many more equations. Furthermore, there are many properties of matrices that can help in solving these kinds of systems.
Multiplying two matrices is only possible when the matrices have the right dimensions. An m times n matrix has to be multiplied with an n times p matrix. The reason for this is that when you multiply two matrices, you have to take the inner product of every row of the first matrix with every column of the second.
This can only be done when both the row vectors of the first matrix and the column vectors of the second matrix have the same length. The result of the multiplication will be an m times p matrix. So it does not matter how many rows A has and columns B has, but the length of the rows of A must be equal to the length of the columns of B.
A special case of matrix multiplication is just multiplying two numbers. This can be seen as a matrix multiplication between two 1x1 matrices. In this case, m, n and p are all equal to 1. Therefore we are allowed to perform the multiplication.
When multiplying two matrices, A and B, we can determine the entries of this multiplication as follows:
When A*B = C, we can determine entry c_i,j by taking the inner product of the i'th row of A with the j'th column of B.
The inner product of two vectors, v and w, is equal to the sum of v_i*w_i for i from 1 to n. Here n is the length of the vectors v and w. An example:
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(2,4,3) • (1,5,7) = 2*1 + 4*5 + 3*7 = 43
Another way to define the inner product of v and w is to describe it as the product of v with the transpose of w. An inner product is always a number. It can never be a vector.
The following picture gives a better understanding of exactly how matrix multiplication works.
In the picture we see that 1*7 + 2*9 + 3* 11 = 58 forms the first entry. The second is determined by taking the inner product of (1,2,3) and (8,10,12), which is 1*8+3*10+3*12 = 64. Then the second row will be 4*7 + 5*9 + 6* 11 = 139 and 4*8 + 5*10 + 6*12 = 154.
As you can see, a 2-times-3 matrix multiplied by a 3-times-2 matrix gives a 2-times-2 square matrix.
Properties of Matrix Multiplication
Matrix multiplication does not have the same properties as normal multiplication. First, we don't have commutativity, which means that A*B does not have to be equal to B*A. This is a general statement. This means that there are matrices for which A*B = B*A, for example, when A and B are just numbers. However, it is not true for any pair of matrices.
However, it satisfies associativity, which means A*(B*C) = (A*B)*C.
It also satisfies distributivity, meaning A(B+C) = AB + AC. This is called left distributivity.
Right distributivity means (B+C)A = BA + CA. This is also satisfied. However, AB + AC is not necessarily equal to BA + CA since matrix multiplication is not commutative.
Special Kinds of Matrices
The first special matrix that comes up is a diagonal matrix. A diagonal matrix is a matrix that has non-zero elements on the diagonal and zero everywhere else. A special diagonal matrix is the identity matrix, mostly denoted as I. This is a diagonal matrix where all diagonal elements are 1. Multiplying any matrix A with the identity matrix, either left or right, results in A, so:
A*I = I*A = A
Another special matrix is the inverse matrix of a matrix A, mostly denoted as A^-1. The special property here is as follows:
A*A^-1 = A^-1*A = I
So multiplying a matrix with its inverse results in the identity matrix.
Not all matrices have an inverse. First of all, a matrix needs to be square to have an inverse. This means that the number of rows is equal to the number of columns, so we have an n x n matrix. But even being square is not enough to guarantee that the matrix has an inverse. A square matrix that does not have an inverse is called a singular matrix; therefore, a matrix with an inverse is called non-singular.
A matrix has an inverse if and only if its determinant is not equal to zero. So any matrix that has a determinant equal to zero is singular, and any square matrix that doesn't have a determinant equal to zero has an inverse.
Different Kinds of Matrix Multiplication
The way described above is the standard way of multiplying matrices. There are some other ways to do it that can be valuable for certain applications. Examples of these different multiplication methods are the Hadamard product and the Kronecker product.
Two matrices A and B can be multiplied if the rows of the first matrix have the same length as the columns of the second matrix. Then the entries of the product can be determined by taking the inner products of the rows of A and the columns of B. Therefore AB is not the same as BA.
The identity matrix I is special in the sense that IA = AI = A. When a matrix A is multiplied with its inverse A^-1 you get the identity matrix I.
This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.