R: Matrix Operations

Matrix manipulation in R are very useful in Linear Algebra. Below are list of common yet important functions in dealing operations with matrices:

• Transpose - t;
• Multiplication - %*%;
• Determinant - det; and,
• Inverse - solve, or ginv of MASS library
• Eigenvalues and Eigenvectors - eigen

Consider these matrices, $\left[\begin{array}{ccc}3&4&5\\2&1&3\\6&5&4\end{array}\right]$ and  $\left[\begin{array}{ccc}6&7&5\\4&5&8\\7&6&6\end{array}\right]$. In R, these would be,

	dat1 <- c(3,4,5,2,1,3,6,5,4)
	matrix1 <- matrix(dat1, nrow = 3, ncol = 3, byrow = TRUE)
	matrix1
	[,1] [,2] [,3]
	[1,] 3 4 5
	[2,] 2 1 3
	[3,] 6 5 4
	dat2 <- c(6,7,5,4,5,8,7,6,6)
	matrix2 <- matrix(dat2, nrow = 3, ncol = 3, byrow = TRUE)
	matrix2
	[,1] [,2] [,3]
	[1,] 6 7 5
	[2,] 4 5 8
	[3,] 7 6 6

view raw MatrixOp1.R hosted with ❤ by GitHub

Transposing these, simply use t

	t(matrix1)
	[,1] [,2] [,3]
	[1,] 3 2 6
	[2,] 4 1 5
	[3,] 5 3 4
	t(matrix2)
	[,1] [,2] [,3]
	[1,] 6 4 7
	[2,] 7 5 6
	[3,] 5 8 6

view raw MatrixOp2.R hosted with ❤ by GitHub

Multiplying these, would be

	matrix1 %*% matrix2
	[,1] [,2] [,3]
	[1,] 69 71 77
	[2,] 37 37 36
	[3,] 84 91 94

view raw MatrixOp3.R hosted with ❤ by GitHub

For the determinant, we have

	det(matrix1)
	[1] 27
	det(matrix2)
	[1] 61

view raw MatrixOp4.R hosted with ❤ by GitHub

Taking the inverse of matrix1 is achieved by solve or ginv R functions. Note that ginv is in MASS package, and so we have

	solve(matrix1)
	[,1] [,2] [,3]
	[1,] -0.4074074 0.3333333 0.25925926
	[2,] 0.3703704 -0.6666667 0.03703704
	[3,] 0.1481481 0.3333333 -0.18518519
	library(MASS)
	ginv(matrix1)
	[,1] [,2] [,3]
	[1,] -0.4074074 0.3333333 0.25925926
	[2,] 0.3703704 -0.6666667 0.03703704
	[3,] 0.1481481 0.3333333 -0.18518519

view raw MatrixOp5.R hosted with ❤ by GitHub

Finally, for eigenvalues and eigenvectors simply use eigen

	eigen(matrix1)
	$values
	[1] 11.238947 -2.088872 -1.150075
	$vectors
	[,1] [,2] [,3]
	[1,] -0.6017346 -0.3458537 -0.4307244
	[2,] -0.3305706 -0.5288292 0.8445517
	[3,] -0.7270754 0.7750644 -0.3181336
	eigen(matrix2)
	$values
	[1] 18.020246+0.000000i -0.510123+1.767726i -0.510123-1.767726i
	$vectors
	[,1] [,2] [,3]
	[1,] 0.5729619+0i -0.4414068-0.3843531i -0.4414068+0.3843531i
	[2,] 0.5495930+0i 0.6888097+0.0000000i 0.6888097+0.0000000i
	[3,] 0.6079985+0i -0.2537249+0.3443799i -0.2537249-0.3443799i

view raw MatrixOp6.R hosted with ❤ by GitHub

The output above returns the $values, which is the eigenvalues, and $vectors, the eigenvectors.