Identity matrix

In linear algebra, the identity matrix of size $n$ is the $n\times n$ square matrix with ones on the main diagonal and zeros elsewhere. It has unique properties, for example when the identity matrix represents a geometric transformation, the object remains unchanged by the transformation. In other contexts, it is analogous to multiplying by the number 1.

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Voiceover:When you first learned multiplication many, many, many years ago, you got exposed to the idea that 1 times ... I shouldn't use that symbol ... 1 times some number is equal to that number again, and that makes intuitive sense. You're just literally saying one of this thing is just going to be that thing right over there. And you could view it as 1, when you're thinking about regular multiplication or scalar multiplication, it has this identity property. It has the identity property of multiplication. 1 times some number is equal to that some number again. Since we're now exploring matrices and matrix multiplication, the question arises is there some matrix that has the same property for matrix multiplication? To make that a little bit more concrete, is there some matrix I, and let me bold it as best as I can in my handwriting, is there some matrix I that if I were to multiply it times any other ... I think I over-bolded that one, but I'll just go with it. If I were to multiply it times any other matrix, A, that the resulting product is going to be matrix A again by the standard conventions of matrix multiplication. To make that a little bit concrete, let's just imagine. Let's just take an example for A. Let's say that our matrix A, let's go 3 by 3. Let's say it is 1, 2, 3, 4, 5, 6, 7, 8, 9. What I encourage you to is pause this video and try to think about whether you can construct some matrix I, and first think about even what the dimensions of matrix I have to be in order to, when you multiply the two this way, when you multiply I times A, you get A again. I'm assuming you've given a go at it, so let's think this through. Let's throw matrix A down there. Let's say copy and paste. Let's first think about what the dimensions are going to have to be. When I multiply my matrix I, when I multiply my matrix I times A right over here, I get A again. I'm multiplying something times a 3 by 3, 3 by 3 matrix, and I'm getting another 3 by 3 matrix. There's a few things that we know. First of all, in order for this matrix multiplication to even be defined, this matrix, the identity matrix, has to have the same number of columns as A has rows. We already see that A has 3 rows, so this character, the identity matrix, is going to have to have 3 columns. It's going to have to have 3 columns. We also know that the dimensions of the product, the rows of the product are defined by the rows of the first matrix, so this has to be also a 3 by 3, and of course, the columns of the product are defined by the columns of the second matrix. This is what defines this. These middle two have to match, and then the rows of the first matrix define the rows of the product, and then the columns of the second matrix define the columns of the product. We know this has to be a 3 by 3 matrix. Now what else do we know? We know what the product needs to be. It also needs to be 1, 2, 3, 4, 5, 6, 7, 8, 9. Let's think about it. To get this first entry right over here, we're going to have to multiply this row, this row times this column, since you take the dot product of it. I'm going to have to multiply something times 1 plus something else times 4 plus something else times 7 to get 1. Let's just think about it in the most, I guess we could say, naive possible way. What happens if we just multiply 1 times this 1 to get 1 and then 0 times 4 and add to it and then 0 times 7. I think that works out. When you take this product, this entry right over here is going to be 1 times 1, 1 times 1 plus 0 times 4, 0 times 4 plus 0 times 7, plus 0 times 7. That worked out quite well, but let's just make sure that that still holds. What happens when we multiply this row times this column or times this column to get this entry right over here? It works out. It's 1 times 2 plus 0 times 5 plus 0 times 8, so it makes sense. You get 2 again. Same thing when you do it for this 3rd column. 1 times 3 plus 0 times 6 plus 0 times 9 is going to be 3. Now what do we do in the second row? Let's think about it a little bit. The second row right over here is going to determine what values we get over here. For example, to get this entry right over there, we're going to multiply this row, we're going to multiply this row times this column, times this column. We want it to have the 4, so one way to think about it, we just want this middle entry here, so let's multiply 0 times 1 plus 1 times 4 plus 0 times 7, and then we're going to get 4. That works out for this next entry right over here. 0 times 2 plus 1 times 5 plus 0 times 8. We get 5. It will work out the same for this entry over there. Now, for this last entry, for this bottom row right over here of our product, to do that, we're going to have to multiply this row times these columns, or take, I guess you could say, the dot product. To get the 7, we want to multiply this row times this column, or take the dot product of this row and that column. If we want the 7, let's multiply 0 times a 1 plus 0 times a 4 plus a 1 times the 7. Just like that, you'll see that that works. That gives us a 7 for this entry. It gives us, when you take the dot of this and that, it gives you an 8 for this entry. You take the dot product of that and that. It gives you the 9, the 9 for that entry. Just like that, we have constructed a 3 by 3 identity matrix. The 3 by 3 identity matrix is equal to 1, 0, 0, 0, 1, 0, and 0, 0, 1. As you will see, whenever you construct an identity matrix, if you're constructing a 2 by 2 identity matrix, so I can say identity matrix 2 by 2, it's going to have a very similar pattern. It's going to be 1, 0, 0, 1. If you have a 4 by 4 identity matrix, it is going to be, you could guess it, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1. You essentially just have 1s down the diagonal going from the top left to the bottom right. What's neat about identity matrices, you multiply it times any matrix, and you're going to get that matrix again. Now another thing I encourage you to do is we've just shown that I times A is equal to A, but I'll let you do this after this video, what about A times I? We've seen that matrix multiplication, the order matters, so what happens here? If you take A times I, do you still get A?

Terminology and notation

The identity matrix is often denoted by $I_{n}$ , or simply by $I$ if the size is immaterial or can be trivially determined by the context.^[1]

I_{1}={\begin{bmatrix}1\end{bmatrix}},\ I_{2}={\begin{bmatrix}1&0\\0&1\end{bmatrix}},\ I_{3}={\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}},\ \dots ,\ I_{n}={\begin{bmatrix}1&0&0&\cdots &0\\0&1&0&\cdots &0\\0&0&1&\cdots &0\\\vdots &\vdots &\vdots &\ddots &\vdots \\0&0&0&\cdots &1\end{bmatrix}}.

The term unit matrix has also been widely used,^[2]^[3]^[4]^[5] but the term identity matrix is now standard.^[6] The term unit matrix is ambiguous, because it is also used for a matrix of ones and for any unit of the ring of all $n\times n$ matrices.^[7]

In some fields, such as group theory or quantum mechanics, the identity matrix is sometimes denoted by a boldface one, $\mathbf {1}$ , or called "id" (short for identity). Less frequently, some mathematics books use $U$ or $E$ to represent the identity matrix, standing for "unit matrix"^[2] and the German word Einheitsmatrix respectively.^[8]

In terms of a notation that is sometimes used to concisely describe diagonal matrices, the identity matrix can be written as

I_{n}=\operatorname {diag} (1,1,\dots ,1).

The identity matrix can also be written using the Kronecker delta notation:^[8]

(I_{n})_{ij}=\delta _{ij}.

Properties

When $A$ is an $m\times n$ matrix, it is a property of matrix multiplication that

I_{m}A=AI_{n}=A.

In particular, the identity matrix serves as the multiplicative identity of the matrix ring of all

n\times n

matrices, and as the identity element of the general linear group

GL(n)

, which consists of all invertible

n\times n

matrices under the matrix multiplication operation. In particular, the identity matrix is invertible. It is an involutory matrix, equal to its own inverse. In this group, two square matrices have the identity matrix as their product exactly when they are the inverses of each other.

When $n\times n$ matrices are used to represent linear transformations from an $n$ -dimensional vector space to itself, the identity matrix $I_{n}$ represents the identity function, for whatever basis was used in this representation.

The $i$ th column of an identity matrix is the unit vector $e_{i}$ , a vector whose $i$ th entry is 1 and 0 elsewhere. The determinant of the identity matrix is 1, and its trace is $n$ .

The identity matrix is the only idempotent matrix with non-zero determinant. That is, it is the only matrix such that:

When multiplied by itself, the result is itself
All of its rows and columns are linearly independent.

The principal square root of an identity matrix is itself, and this is its only positive-definite square root. However, every identity matrix with at least two rows and columns has an infinitude of symmetric square roots.^[9]

The rank of an identity matrix $I_{n}$ equals the size $n$ , i.e.:

\operatorname {rank} (I_{n})=n.

Notes

^ "Identity matrix: intro to identity matrices (article)". Khan Academy. Retrieved 2020-08-14.
^ ^a ^b Pipes, Louis Albert (1963). Matrix Methods for Engineering. Prentice-Hall International Series in Applied Mathematics. Prentice-Hall. p. 91.
^ Roger Godement, Algebra, 1968.
^ ISO 80000-2:2009.
^ Ken Stroud, Engineering Mathematics, 2013.
^ ISO 80000-2:2019.
^ Weisstein, Eric W. "Unit Matrix". mathworld.wolfram.com. Retrieved 2021-05-05.
^ ^a ^b Weisstein, Eric W. "Identity Matrix". mathworld.wolfram.com. Retrieved 2020-08-14.
^ Mitchell, Douglas W. (November 2003). "87.57 Using Pythagorean triples to generate square roots of $I_{2}$ ". The Mathematical Gazette. 87 (510): 499–500. doi:10.1017/S0025557200173723. JSTOR 3621289.

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This page was last edited on 13 August 2023, at 00:48

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Terminology and notation

Properties

See also

Notes