Inverse of a matrix

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The inverse of a matrix plays the same roles in matrix algebra as the reciprocal of a number and division does in ordinary arithmetic: Just as we can solve a simple equation like $4 x = 8$ for $x$ by multiplying both sides by the reciprocal $$ 4 x = 8 \Rightarrow 4^{-1} 4 x = 4^{-1} 8 \Rightarrow x = 8 / 4 = 2$$ we can solve a matrix equation like $\mathbf{A x} = \mathbf{b}$ for the vector $\mathbf{x}$ by multiplying both sides by the inverse of the matrix $\mathbf{A}$, $$\mathbf{A x} = \mathbf{b} \Rightarrow \mathbf{A}^{-1} \mathbf{A x} = \mathbf{A}^{-1} \mathbf{b} \Rightarrow \mathbf{x} = \mathbf{A}^{-1} \mathbf{b}$$

The following examples illustrate the basic properties of the inverse of a matrix.

Load the matlib package

This defines: inv(), Inverse(); the standard R function for matrix inverse is solve()

library(matlib)

Create a 3 x 3 matrix

The ordinary inverse is defined only for square matrices.

A <- matrix( c(5, 1, 0, 3,-1, 2, 4, 0,-1), nrow=3, byrow=TRUE) det(A)

Basic properties

1. det(A) != 0, so inverse exists

Only non-singular matrices have an inverse.

(AI <- inv(A))

2. Definition of the inverse: $A^{-1} A = A A^{-1} = I$ or AI * A = diag(nrow(A))

The inverse of a matrix $A$ is defined as the matrix $A^{-1}$ which multiplies $A$ to give the identity matrix, just as, for a scalar $a$, $a a^{-1} = a / a = 1$.

NB: Sometimes you will get very tiny off-diagonal values (like 1.341e-13). The function zapsmall() will round those to 0.

AI %*% A

3. Inverse is reflexive: inv(inv(A)) = A

Taking the inverse twice gets you back to where you started.

inv(AI)

4. inv(A) is symmetric if and only if A is symmetric

inv( t(A) ) is_symmetric_matrix(A) is_symmetric_matrix( inv( t(A) ) )

Here is a symmetric case:

B <- matrix( c(4, 2, 2, 2, 3, 1, 2, 1, 3), nrow=3, byrow=TRUE) inv(B) inv( t(B) ) is_symmetric_matrix(B) is_symmetric_matrix( inv( t(B) ) ) all.equal( inv(B), inv( t(B) ) )

More properties of matrix inverse

1. inverse of diagonal matrix = diag( 1/ diagonal)

In these simple examples, it is often useful to show the results of matrix calculations as fractions, using MASS::fractions().

D <- diag(c(1, 2, 4)) inv(D) MASS::fractions( diag(1 / c(1, 2, 4)) )

2. Inverse of an inverse: inv(inv(A)) = A

A <- matrix(c(1, 2, 3, 2, 3, 0, 0, 1, 2), nrow=3, byrow=TRUE) AI <- inv(A) inv(AI)

3. inverse of a transpose: inv(t(A)) = t(inv(A))

inv( t(A) ) t( inv(A) )

4. inverse of a scalar matrix: `inv( kA ) = (1/k) * inv(A)`

inv(5 * A) (1/5) * inv(A)

5. inverse of a matrix product: inv(A * B) = inv(B) %*% inv(A)

B <- matrix(c(1, 2, 3, 1, 3, 2, 2, 4, 1), nrow=3, byrow=TRUE) C <- B[, 3:1] A %*% B inv(A %*% B) inv(B) %*% inv(A)

This extends to any number of terms: the inverse of a product is the product of the inverses in reverse order.

(ABC <- A %*% B %*% C) inv(A %*% B %*% C) inv(C) %*% inv(B) %*% inv(A) inv(ABC)

6. $\det (A^{-1}) = 1 / \det(A) = [\det(A)]^{-1}$

The determinant of an inverse is the inverse (reciprocal) of the determinant

det(AI) 1 / det(A)

Geometric interpretations

Some of these properties of the matrix inverse can be more easily understood from geometric diagrams. Here, we take a $2 \times 2$ non-singular matrix $A$,

A <- matrix(c(2, 1, 1, 2), nrow=2, byrow=TRUE) A det(A)

The larger the determinant of $A$, the smaller is the determinant of $A^{-1}$.

AI <- inv(A) MASS::fractions(AI) det(AI)

Now, plot the rows of $A$ as vectors $a_1, a_2$ from the origin in a 2D space. As illustrated in vignette("det-ex1"), the area of the parallelogram defined by these vectors is the determinant.

par(mar=c(3,3,1,1)+.1) xlim <- c(-1,3) ylim <- c(-1,3) plot(xlim, ylim, type="n", xlab="X1", ylab="X2", asp=1) sum <- A[1,] + A[2,] # draw the parallelogram determined by the rows of A polygon( rbind(c(0,0), A[1,], sum, A[2,]), col=rgb(1,0,0,.2)) vectors(A, labels=c(expression(a[1]), expression(a[2])), pos.lab=c(4,2)) vectors(sum, origin=A[1,], col="gray") vectors(sum, origin=A[2,], col="gray") text(mean(A[,1]), mean(A[,2]), "A", cex=1.5)

The rows of the inverse $A^{-1}$ can be shown as vectors $a^1, a^2$ from the origin in the same space.

par(mar=c(3,3,1,1)+.1) xlim <- c(-1,3) ylim <- c(-1,3) plot(xlim, ylim, type="n", xlab="X1", ylab="X2", asp=1) sum <- A[1,] + A[2,] # draw the parallelogram determined by the rows of A polygon( rbind(c(0,0), A[1,], sum, A[2,]), col=rgb(1,0,0,.2)) vectors(A, labels=c(expression(a[1]), expression(a[2])), pos.lab=c(4,2)) vectors(sum, origin=A[1,], col="gray") vectors(sum, origin=A[2,], col="gray") text(mean(A[,1]), mean(A[,2]), "A", cex=1.5) vectors(AI, labels=c(expression(a^1), expression(a^2)), pos.lab=c(4,2)) sum <- AI[1,] + AI[2,] polygon( rbind(c(0,0), AI[1,], sum, AI[2,]), col=rgb(0,0,1,.2)) text(mean(AI[,1])-.3, mean(AI[,2])-.2, expression(A^{-1}), cex=1.5)

Thus, we can see:

  • The shape of $A^{-1}$ is a $90^o$ rotation of the shape of $A$.

  • $A^{-1}$ is small in the directions where $A$ is large.

  • The vector $a^2$ is at right angles to $a_1$ and $a^1$ is at right angles to $a_2$

  • If we multiplied $A$ by a constant $k$ to make its determinant larger (by a factor of $k^2$), the inverse would have to be divided by the same factor to preserve $A A^{-1} = I$.

One might wonder whether these properties depend on symmetry of $A$, so here is another example, for the matrix A <- matrix(c(2, 1, 1, 1), nrow=2), where $\det(A)=1$.

(A <- matrix(c(2, 1, 1, 1), nrow=2)) (AI <- inv(A))

The areas of the two parallelograms are the same because $\det(A) = \det(A^{-1}) = 1$.

par(mar=c(3,3,1,1)+.1) xlim <- c(-1,3) ylim <- c(-1,3) plot(xlim, ylim, type="n", xlab="X1", ylab="X2", asp=1) sum <- A[1,] + A[2,] # draw the parallelogram determined by the rows of A polygon( rbind(c(0,0), A[1,], sum, A[2,]), col=rgb(1,0,0,.2)) vectors(A, labels=c(expression(a[1]), expression(a[2])), pos.lab=c(4,2)) vectors(sum, origin=A[1,], col="gray") vectors(sum, origin=A[2,], col="gray") text(mean(A[,1]), mean(A[,2]), "A", cex=1.5) vectors(AI, labels=c(expression(a^1), expression(a^2)), pos.lab=c(4,2)) sum <- AI[1,] + AI[2,] polygon( rbind(c(0,0), AI[1,], sum, AI[2,]), col=rgb(0,0,1,.2)) text(-.1, -.1, expression(A^{-1}), cex=1.5)