If in the multiplication, the identity matrix is the first factor, then the identity matrix must have dimensions with as many columns as the matrix it is multiplying has rows.Įquation 6: Multiplying a matrix by its inverse.
An identity matrix is capable of multiplying any matrix with any order (dimensions) as long as it follows the next rules:.
An identity matrix is always an square matrix:Īs seen in equations 1 and 2, the order of an identity matrix is always n, which refers to the dimensions nxn (meaning there is always the same amount of rows and columns in the matrix).
Notice that since an identity matrix is a square matrix, it contains the same amount of rows and columns and so, its order can be referred simply as nxn, or as a one digit subindex as shown in equations 1 and 2, which we call "n" (given that both m and n dimensions coming from the typical matrix notation are equal for square matrices, only one of those letters is necessary as a subindex to describe the order of such matrices).Īnd so, in equation 2 we can easily see that I2 refers to an identity matrix with two rows and two columns, which at the same time has only two elements in its main diagonal a notation of I3 corresponds to an identity matrix of order 3, or one containing three rows and three columns and 3 elements on its main diagonal and the notation system continues that way for any subindex n. Remember that the order of a matrix refers to the amount of rows and columns it contains, which are also called its dimensions mxn. Mathematically, the identity matrix is represented as:Įquation 2: Examples of identity matrices of different dimensions In the next section we will take a look into the properties of the identity matrix, and the unit matrix definition will make much more sense, especially in the case of matrix multiplications including an identity matrix (look at property number 3). In other words, the identity matrix is the equivalent to the unit of one, but in this case it happens to be an algebraic object with dimensions and array organization which can be used in operations with other ordered number arrays (other matrices). In this case, all of the non-zero entries in the matrix will have a value of one, and that happens to be one of the reasons why the identity matrix is sometimes called the unit matrix too.įor the case of matrix linear algebra notation, the identity matrix serves as the equivalent object to the unit in numerical algebra (other reason why is called the unit matrix). Given the characteristics of an identity matrix, we can also conclude these type of matrices are also diagonal matrices.Ī diagonal matrix is that in which all of its element entries are equal to zero, except for the elements found on its main diagonal. The order of a matrix comes from its dimensions, and its main diagonal refers to the array of elements inside the matrix which form an inclined line from the top left corner to the bottom right corner. To explain part by part this definition, let us start by reminding you that a square matrix refers to a matrix containing the same amount of rows and columns. What is an identity matrix?Īn identity matrix is a given square matrix of any order which contains on its main diagonal elements with value of one, while the rest of the matrix elements are equal to zero. Having learned about the zero matrix, it is time to study another type of matrix containing a constant specific set of values every time, is time for us to study the identity matrices.