Identity matrix

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In linear algebra, the matrix identity is a matrix that fulfills the property of being the neutral element of the matrices product. This means that the product of any matrix by the identity matrix (where the product is defined) has no effect. The column i-that of an identity matrix is the unit vector ${displaystyle e_{i},}$ of a vector base immersed in an Euclid Space n. Each matrix represents a linear application between two finite dimension vector spaces. La matrix identity it is so called because it represents the identity application that goes from a finite dimension vector space to itself.

Definition

Since the product of matrices only makes sense if its dimensions are compatible, there are infinite identity matrices depending on the dimensions. ${displaystyle I_{n},}$ , the size identity matrix $n,$ , is defined as of the entrances of the main diagonal: ${displaystyle I_{1}={begin{pmatrix}1\end{pmatrix}}, I_{2}={begin{pmatrix}1&0\0&1\end{pmatrix}}, I_{3}={begin{pmatrix}1&0&0\0&1&0\0&0&1\end{pmatrix}}, cdots I_{n}={begin{pmatrix}1&0&cdots &0\0&1&cdots &0\vdots &vdots &ddots &vdots \0&0&cdots &1\end{pmatrix}}}$

Using the notation sometimes used to concisely describe diagonal matrices, it turns out:

{displaystyle I_{n}=mathrm {diag} (1,1,...,1),}

If the size is immaterial, or can be deducted trivially by the context, then it is simply written as ${displaystyle I,}$ .

Can also be written using Kronecker delta notation:

{displaystyle I_{ij}=delta _{ij},}

or, even more simply,

{displaystyle I=(delta _{ij}),}

The identity matrix of order n can also be considered as the permutation matrix that is the neutral element of the group of permutation matrices of order n!.

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