Matrix

Matrices are a two-dimensional arrangement of numbers, a certain number of rows and a certain number of columns. Matrices are used for multiple applications and generally represent the coefficients of systems of linear equations or to represent linear applications; in this case matrices play the same role as vector data for linear maps. They can be added, multiplied, and broken down in various ways, which also makes them a key concept in field of linear algebra.

\[\begin{equation*} A_{(row , column)} = A_{(n , k)} \begin{pmatrix} a_{11} & a_{12} & \dots & a_{1k} \\ a_{21} & a_{22} & \dots & a_{2k} \\ \vdots & \vdots & \ddots & \vdots\\ a_{n1} & a_{n2} & \dots & a_{nk} \end{pmatrix} \end{equation*}\]

x <-c(2,3,5,-5,3,3,-9,-2,0,8,-2,3)
M <-matrix(x,nrow=4,ncol=3)
M

##      [,1] [,2] [,3]
## [1,]    2    3    0
## [2,]    3    3    8
## [3,]    5   -9   -2
## [4,]   -5   -2    3

#Creating from vectors:
age <- c(28,25,32,27,36,23)
height <- c(170,175,180,165,169,175)
weight <- c(75,80,95,80,75,68)
Z <-matrix(c(age,height,weight), nrow=6,ncol=3)
Z

##      [,1] [,2] [,3]
## [1,]   28  170   75
## [2,]   25  175   80
## [3,]   32  180   95
## [4,]   27  165   80
## [5,]   36  169   75
## [6,]   23  175   68

#We can add the names or labels of rows or columns
colnames(Z) <- c('age','height','weight')
rownames(Z) <- c('John','Peter','Andrea','Melissa','Sara','Julio')
Z

##         age height weight
## John     28    170     75
## Peter    25    175     80
## Andrea   32    180     95
## Melissa  27    165     80
## Sara     36    169     75
## Julio    23    175     68

Matrix Types

Square Matrix

A square matrix is any matrix that has the same number of rows and columns. \[\begin{equation*} A_{(3,3)} = \begin{pmatrix} 7\hspace{0.5cm}6\hspace{0.5cm}0\\ 5\hspace{0.5cm}3\hspace{0.5cm}1\\ 2\hspace{0.5cm}2\hspace{0.5cm}1\\ \end{pmatrix} \end{equation*}\]

x <-c(7,5,2,6,3,2,0,1,1)
A <-matrix(x,nrow=3,ncol=3)
A

##      [,1] [,2] [,3]
## [1,]    7    6    0
## [2,]    5    3    1
## [3,]    2    2    1

Symmetric Matrix

A symmetric matrix is a matrix of order n with the same number of rows and columns where its transposed matrix is equal to the original matrix. \[ a_{(i,k)}=a_{(k,i)} \]

\[\begin{equation*} A_{(3,3)} = \begin{pmatrix} a_{11} \hspace{0.2cm} a_{12}\hspace{0.2cm} a_{13} \\ a_{21} \hspace{0.2cm}a_{22}\hspace{0.2cm} a_{23} \\ a_{31} \hspace{0.2cm}a_{32}\hspace{0.2cm} a_{33} \\ \end{pmatrix} \hspace{2.5cm} B_{(3,3)} = \begin{pmatrix} 5\hspace{0.2cm} 7\hspace{0.2cm} 10 \\ 7 \hspace{0.2cm}9 \hspace{0.2cm}8 \\ 10\hspace{0.2cm} 8 \hspace{0.2cm}15 \\ \end{pmatrix} \end{equation*}\]

x <-c(1,2,7,2,8,1,7,1,9)
A <-matrix(x,nrow=3,ncol=3)
A

##      [,1] [,2] [,3]
## [1,]    1    2    7
## [2,]    2    8    1
## [3,]    7    1    9

Diagonal Matrix

In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero.

\[\begin{equation*} \Large{D_{(3,3)}=}\begin{bmatrix} 7 \hspace{0.2cm} 0\hspace{0.2cm} 0 \\ 0 \hspace{0.2cm}6\hspace{0.2cm} 0 \\ 0 \hspace{0.2cm}0\hspace{0.2cm} 8 \\ \end{bmatrix} \end{equation*}\]

C1 <- c(7,0,0)
C2 <- c(0,6,0)
C3 <- c(0,0,8)
D <-matrix(c(C1,C2,C3), nrow=3,ncol=3)
D

##      [,1] [,2] [,3]
## [1,]    7    0    0
## [2,]    0    6    0
## [3,]    0    0    8

Scalar Matrix

The scalar matrix is a square matrix having a constant value for all the elements of the principal diagonal, and the other elements of the matrix are zero. The scalar matrix is obtained by the product of the identity matrix with a numeric constant value.

\[\begin{equation*} \Large{E_{(3,3)}=}\begin{bmatrix} 7 \hspace{0.2cm} 0\hspace{0.2cm} 0 \\ 0 \hspace{0.2cm}7\hspace{0.2cm} 0 \\ 0 \hspace{0.2cm}0\hspace{0.2cm} 7 \\ \end{bmatrix} \end{equation*}\]

C1 <- c(7,0,0)
C2 <- c(0,7,0)
C3 <- c(0,0,7)
E <-matrix(c(C1,C2,C3), nrow=3,ncol=3)
E

##      [,1] [,2] [,3]
## [1,]    7    0    0
## [2,]    0    7    0
## [3,]    0    0    7

Identity Matrix

An 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.

\[\begin{equation*} \Large{I_{(3)}=}\begin{bmatrix} 1 \hspace{0.2cm} 0\hspace{0.2cm} 0 \\ 0 \hspace{0.2cm}1\hspace{0.2cm} 0 \\ 0 \hspace{0.2cm}0\hspace{0.2cm} 1 \\ \end{bmatrix} \end{equation*}\]

x <-c(1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1)
I <-matrix(x,nrow=4,ncol=4)
I

##      [,1] [,2] [,3] [,4]
## [1,]    1    0    0    0
## [2,]    0    1    0    0
## [3,]    0    0    1    0
## [4,]    0    0    0    1

Triangle Matrix

A triangular matrix is a special type of square matrix where all the values above or below the diagonal are zero. L is called a lower triangular matrix and U is called an upper triangular matrix. Matrix equations of above form can be easily solved using backward substitution or forward substitution.

\[\begin{equation*} \Large{P_{(3,3)}=}\begin{bmatrix} 2 \hspace{0.2cm} \color{red}0\hspace{0.2cm} \color{red}0 \\ 8 \hspace{0.2cm}7\hspace{0.2cm} \color{red}0 \\ 6 \hspace{0.2cm}9\hspace{0.2cm} 7 \end{bmatrix} \quad \text{(Lower Triangular Matrix)} \end{equation*}\]\

\[\begin{equation*} \Large{Q_{(3,3)}=}\begin{bmatrix} 4 \hspace{0.2cm} 1\hspace{0.2cm} 6 \\ \color{red}0 \hspace{0.2cm} 5 \hspace{0.2cm} 8\\ \color{red}0 \hspace{0.2cm} \color{red}0 \hspace{0.2cm} 9 \\ \end{bmatrix} \quad \text{(Upper Triangular Matrix)} \end{equation*}\]

Matrix Operations

For two matrices to be equal, they must have the same number of columns and rows, since each element must be equal. Let matrix A equal matrix B if: \(a_{(i,k)}=b_{(i,k)}\) \[\begin{equation*} A_{(2,2)}= \begin{bmatrix} 5\hspace{0.5cm} 7\\ 7 \hspace{0.5cm}9 \\ \end{bmatrix}\hspace{2cm} = \hspace{2cm} B_{(2,2)}=\begin{bmatrix} 5\hspace{0.5cm} 7\\ 7 \hspace{0.5cm}9 \\ \end{bmatrix} \end{equation*}\]

Transposed Matrix

The transpose of a matrix is simply a flipped version of the original matrix. We can transpose a matrix by switching its rows with its columns. \(a_{(i,k)}\) = \(a'_{(k,i)}\).

\[\begin{equation*} A_{(2,3)} \hspace{2cm} ; \hspace{2cm} A'_{(3,2)} \end{equation*}\]

\[\begin{equation*} \Large{A_{(2,3)}=} \begin{bmatrix} \color{red}a_{11} \hspace{0.2cm} \color{red}b_{12}\hspace{0.2cm} \color{red}c_{13} \\ \color{blue}d_{21} \hspace{0.2cm}\color{blue}e_{22}\hspace{0.2cm} \color{blue}f_{23} \end{bmatrix} \Large{A'_{(3,2)}=}\begin{bmatrix} \color{red}a_{11} \hspace{0.2cm} \color{blue}d_{12}\\ \color{red}b_{21} \hspace{0.2cm} \color{blue}e_{22}\\ \color{red}c_{31}\hspace{0.2cm} \color{blue}f_{32} \end{bmatrix} \end{equation*}\]

a <-c(2,5,8,6,2,3)
A <-matrix(a,nrow=2,ncol=3)
A

##      [,1] [,2] [,3]
## [1,]    2    8    2
## [2,]    5    6    3

t(A)

##      [,1] [,2]
## [1,]    2    5
## [2,]    8    6
## [3,]    2    3

Matrix Addition and Subtraction

To be able to add or subtract matrices, they must have the same number of rows and columns. This is so since, for both addition and subtraction, the terms that occupy the same place in the matrices are added or subtracted.

\[\begin{equation*} \Large{A_{(i,k)} + B_{(i,k)} =C_{(i,k)}}\\ \Large{A_{(i,k)} - B_{(i,k)} =D_{(i,k)}}\\ \end{equation*}\]

a <-c(1,2,5,8,8,3)
A <-matrix(a,nrow=2,ncol=3)
b <-c(3,2,2,5,1,0)
B <-matrix(b,nrow=2,ncol=3)
A

##      [,1] [,2] [,3]
## [1,]    1    5    8
## [2,]    2    8    3

##      [,1] [,2] [,3]
## [1,]    3    2    1
## [2,]    2    5    0

A+B

##      [,1] [,2] [,3]
## [1,]    4    7    9
## [2,]    4   13    3

A-B

##      [,1] [,2] [,3]
## [1,]   -2    3    7
## [2,]    0    3    3

Matrix Product

The condition for multiplying matrices is that the first one must have the same number of columns as the second rows. The matrix resulting from the product will have the same number of rows as the first and the same number of columns as the second. \[ A_{({\color{blue}i},\color{red}j)}*B_{({\color{red}j},{\color{blue}n})}=C_{({\color{blue}i,n})} \] \[\begin{equation*} \Large{A_{({\color{red}2},{\color{blue}3})}=}\begin{bmatrix}\ \color{magenta}4 \hspace{0.2cm} \color{magenta}5\hspace{0.2cm} \color{magenta}2 \\ \color{purple}2 \hspace{0.2cm}\color{purple}3\hspace{0.2cm} \color{purple}1 \\ \end{bmatrix} \hspace{3cm} \Large{B_{{(\color{blue}3},{\color{red}2})}=} \begin{bmatrix} \color{green}2 \hspace{0.2cm} \color{brown}4\\ \color{green}7 \hspace{0.2cm} \color{brown}1\\ \color{green}3\hspace{0.2cm} \color{brown}5 \end{bmatrix} \end{equation*}\]\end{equation*} \[\begin{equation*} \Large{AB_{{\color{blue}(2,2)}}=} \begin{bmatrix} {\color{magenta}4} *{\color{green}2} + {\color{magenta}5} * {\color{green}7} + {\color{magenta}2} * {\color{green}3} _{(1,1)} \hspace{2cm} {\color{magenta}4} *{\color{brown}4} + {\color{magenta}5} * {\color{brown}1} + {\color{magenta}2} *{\color{brown}5} _{(1,2)} \\ {\color{purple}2} *{\color{green}2} + {\color{purple}3} * {\color{green}7} + {\color{purple}1} * {\color{green}3}_{(2,1)} \hspace{2cm} {\color{purple}2} *{\color{brown}4} + {\color{purple}3} * {\color{brown}1} + {\color{purple}1} * {\color{brown}5} _{(2,2)} \end{bmatrix} \end{equation*}\] \[\begin{equation*} {AB_{{\color{blue}(2,2)}}=}\begin{bmatrix} 49 \hspace{2cm} 31\\ 28\hspace{2cm}16 \end{bmatrix} \end{equation*}\]

a <-c(4,2,5,3,2,1)
A <-matrix(a,nrow=2,ncol=3)
b <-c(2,7,3,4,1,5)
B <-matrix(b,nrow=3,ncol=2)

A%*%B

##      [,1] [,2]
## [1,]   49   31
## [2,]   28   16

Internal Product

let Y a vector \(nx1\) \[\begin{equation*} \Large{Y_{(n,1)}=} \begin{bmatrix} Y_{1,1}\\ Y_{2,1} \\ \vdots\\ Y_{n,1} \end{bmatrix} \Large{Y'_{(1,n)}=}\begin{bmatrix} Y_{1,1} & Y_{1,2} & \dots & Y_{1,n} \end{bmatrix} \end{equation*}\]

\[\begin{equation*} Y'_{({\color{red}1},n)} Y_{(n,{\color{red}1})} = (Y_{1,1}'*Y_{1,1}) + (Y_{1,2}'*Y_{2,1}) + \dots + (Y_{n,1}'*Y_{1,n}) \\ Y'Y_{({\color{red}1,\color{red}1})}= Y_{1}^2 + Y_{2}^2 + \dots + Y_{n}^2 \\ Y'Y_{({\color{red}1,\color{red}1})}=\sum^{n}_{{i=1}} Y_i^2 \end{equation*}\]

y <-c(5,3,4,9,6,2,2)
Y <- cbind(y)
Y

##      y
## [1,] 5
## [2,] 3
## [3,] 4
## [4,] 9
## [5,] 6
## [6,] 2
## [7,] 2

t(Y)%*%Y

##     y
## y 175

External Product

\[\begin{equation*} \LARGE{Y_{({\color{red}n},1)}}Y'_{(1,{\color{red}n})} = YY'_{({\color{red}n,\color{red}n})} = \begin{bmatrix} {\color{red}Y_{1}^2} \hspace{1cm} Y_{1}Y_{2} \hspace{1cm} \dots \hspace{1cm} Y_{1}Y_{n} \\ Y_{2}Y_{1} \hspace{1cm} {\color{red}Y_{2}^2} \hspace{1cm} \dots \hspace{1cm} Y_{2}Y_{n} \\ \vdots \hspace{2.5cm} \vdots \hspace{1.5cm} \ddots \hspace{1.8cm} \vdots\\ Y_{n}Y_{1} \hspace{1cm} Y_{n}Y_{2} \hspace{1cm} \dots \hspace{1cm} {\color{red}Y_{n}^2} \end{bmatrix} \end{equation*}\]

y <-c(5,3,4,9,6,2,2)
Y <- cbind(y)
Y

##      y
## [1,] 5
## [2,] 3
## [3,] 4
## [4,] 9
## [5,] 6
## [6,] 2
## [7,] 2

Y%*%t(Y)

##      [,1] [,2] [,3] [,4] [,5] [,6] [,7]
## [1,]   25   15   20   45   30   10   10
## [2,]   15    9   12   27   18    6    6
## [3,]   20   12   16   36   24    8    8
## [4,]   45   27   36   81   54   18   18
## [5,]   30   18   24   54   36   12   12
## [6,]   10    6    8   18   12    4    4
## [7,]   10    6    8   18   12    4    4

Sum of Elements

let the vector \(i\):

\[\begin{equation*} \Large{i_{(n,1)}=} \begin{bmatrix} 1 \\ 1 \\ \vdots\\ 1 \end{bmatrix} \hspace{0.5cm} \LARGE{i'_{(1 , n)}} = \begin{bmatrix} 1 & 1 & \dots 1 \end{bmatrix} \end{equation*}\]

To find the sum of elements of the column vector Y:

\[\begin{equation*} \LARGE{Y_{(n , 1)}} = \begin{bmatrix} y_{1} \\ y_{2}\\ \vdots\\ y_{n} \end{bmatrix} \end{equation*}\]

\[\begin{equation*} \Large{i'_{({\color{red}1},n)} Y_{(n,{\color{red}1})} = (y_{1}*1) + (y_{2}*1) + \dots + (y_{n}*1) =\sum^{n}_{{i=1}} Y_i } \end{equation*}\]

y<- c(2,3,5,8,4,3,2,1)
i<- matrix(1,nrow=8,ncol=1)
Y<- cbind(y)
Y

##      y
## [1,] 2
## [2,] 3
## [3,] 5
## [4,] 8
## [5,] 4
## [6,] 3
## [7,] 2
## [8,] 1

##      [,1]
## [1,]    1
## [2,]    1
## [3,]    1
## [4,]    1
## [5,]    1
## [6,]    1
## [7,]    1
## [8,]    1

t(i)%*%Y

##       y
## [1,] 28

#Another way to find the sum of elements of a vector:
sum(Y)

## [1] 28

Idempotent Matrix

Deviations from the Mean:

Remember \(\bar{x}\):

\[\begin{equation*} i\bar{x}= i \frac{1}{n}i'x \end{equation*}\]

\[\begin{equation*} \begin{bmatrix} \bar{x}\\ \bar{x}\\ \vdots \\ \bar{x} \end{bmatrix} = \underbrace{{\frac{1}{n}} \;i\;i'}_{MATRIZ_{(n,n)}}\;x \end{equation*}\]

\[\begin{equation*} [x-\bar{x}] = [x- {\frac{1}{n}}ii'x] \end{equation*}\]

Remember \([X]=I[X]\)

\[\begin{equation*} [IX - {\frac{1}{n}}ii'X] \end{equation*}\]

\[\begin{equation*} [I - {\frac{1}{n}}ii']X \\ M^ºX \end{equation*}\] The idempotent Matrix has as diagonal \(1-1/n\) and off the diagonal \(-1/n\)

i <-matrix(1,nrow=3,ncol=1) 
Identity <-c(1,0,0,0,1,0,0,0,1)  
I <-matrix(Identity,nrow=3,ncol=3)
iit= i%*%t(i) / 3
Mº= (I-iit)
Mº # Idempotent Matrix

##            [,1]       [,2]       [,3]
## [1,]  0.6666667 -0.3333333 -0.3333333
## [2,] -0.3333333  0.6666667 -0.3333333
## [3,] -0.3333333 -0.3333333  0.6666667

i  # Matrix of ones

##      [,1]
## [1,]    1
## [2,]    1
## [3,]    1

I  # Identity Matrix

##      [,1] [,2] [,3]
## [1,]    1    0    0
## [2,]    0    1    0
## [3,]    0    0    1

Properties:

Mº is symmetric.

MºMº = Mº

Mº’Mº = Mº

For a variable X the sum of squares of deviations is: \[\begin{equation*} \Large{ \sum (X_{i}-\bar{X})^2 = \sum (X_{i}^2 - 2 \bar{X}X_{i}+\bar{X}^2) = (\sum X_{i}^2) - n\bar{X}^2 } \end{equation*}\]

In matrix terms: \[\begin{equation*} \sum (X_{i}-\bar{X})^2 = (X_{i}-\bar{X}) ' (X_{i}-\bar{X}) \\ \end{equation*}\] \[\begin{equation*} (M^{\circ}X)'(M^{\circ}X) \end{equation*}\]

\[\begin{equation*} X'Mº'MºX \end{equation*}\]

\[\begin{equation*} X' M^{\circ}X \end{equation*}\]

x<-c(8,4,6)
X<- cbind(x)
XtMX = t(X)%*%Mº%*%X
XtMX

##   x
## x 8

We can build a Matrix of Sum of Squares and Cross Products of deviations from the means for two vectors \(X\) and \(Y\).

\[\begin{equation*} \sum(X_{i}-\bar{X}) (Y_{i}-\bar{Y})=(M^{\circ}X)'(M^{\circ}Y) \end {equation*}\]

\[\begin{bmatrix} \sum(X_{i}-\bar{X})^2 \hspace{2cm} \sum(X_{i}-\bar{X})(Y_{i}-\bar{Y})\\ \sum(Y_{i}-\bar{Y})(X_{i}-\bar{X})\hspace{2cm} \sum(Y_{i}-\bar{Y})^2 \end{bmatrix}\] \[\begin{bmatrix} X'M^{\circ}X \hspace{2cm} X'M^{\circ}Y\\ Y'M^{\circ}X\hspace{2cm} Y'M^{\circ}Y \end{bmatrix}\]

y <-c(2,3,4)
Y <-cbind(y)

a11 <- t(X)%*%Mº%*%X
a21 <- t(Y)%*%Mº%*%X
a12 <- t(X)%*%Mº%*%Y
a22 <- t(Y)%*%Mº%*%Y

spc <- c(a11,a21,a12,a22)
SPC <- matrix(spc,nrow = 2, ncol = 2 )
SPC

##      [,1] [,2]
## [1,]    8   -2
## [2,]   -2    2

Determinant and Inverse of a Matrix

The use of determinants greatly simplifies the resolution of systems of linear equations. To do this, general properties are applied that allow the discussion and resolution of such systems to be undertaken through a rigorous procedure. It is a matrix inversibility test, for which we work with square matrices.

a <- c(1,2,8,6,1,5,2,3,8,4,9,5,2,0,7,4)
A <- matrix(a,nrow = 4, ncol=4)
A

##      [,1] [,2] [,3] [,4]
## [1,]    1    1    8    2
## [2,]    2    5    4    0
## [3,]    8    2    9    7
## [4,]    6    3    5    4

det(A) # Determinant of Matrix A

## [1] 39

solve(A) # Inverse de of Matrix A

##            [,1]       [,2]       [,3]      [,4]
## [1,]  1.4615385 -1.8205128 -3.0256410  4.564103
## [2,] -1.0769231  1.3589744  1.9487179 -2.871795
## [3,]  0.6153846 -0.5384615 -0.9230769  1.307692
## [4,] -2.1538462  2.3846154  4.2307692 -6.076923

Resume

# Resume and other functions

#We can create a matrix through the union of column vectors
a1 <- c(3,2,5,8,6)
a2 <- c(8,7,3,9,3)
a3 <- c(3,4,1,6,6)
a4 <- c(9,3,1,4,0)
a5 <- c(1,2,0,8,2)

A <-cbind(a1,a2,a3,a4,a5)
A

##      a1 a2 a3 a4 a5
## [1,]  3  8  3  9  1
## [2,]  2  7  4  3  2
## [3,]  5  3  1  1  0
## [4,]  8  9  6  4  8
## [5,]  6  3  6  0  2

#We can create a matrix through the union of row vectors

b1 <- c(0,1,0,1,3)
b2 <- c(1,7,2,2,1)
b3 <- c(2,9,3,5,4)
b4 <- c(9,5,3,4,7)
b5 <- c(6,2,7,6,8)

B <-rbind(b1,b2,b3,b4,b5)
B

##    [,1] [,2] [,3] [,4] [,5]
## b1    0    1    0    1    3
## b2    1    7    2    2    1
## b3    2    9    3    5    4
## b4    9    5    3    4    7
## b5    6    2    7    6    8

dim(A)      # Matrix A Dimension

## [1] 5 5

nrow(A)     # Number of rows in Matrix A

## [1] 5

ncol(A)     # Number of columns in Matrix A

## [1] 5

t(A)        # Transpose of Matrix A

##    [,1] [,2] [,3] [,4] [,5]
## a1    3    2    5    8    6
## a2    8    7    3    9    3
## a3    3    4    1    6    6
## a4    9    3    1    4    0
## a5    1    2    0    8    2

solve(A)    # Inverse of Matrix A

##           [,1]        [,2]       [,3]        [,4]        [,5]
## a1  0.01483680 -0.13395507  0.1708351  0.02437474  0.02903773
## a2 -0.10089021  0.28232302  0.1526070 -0.02289106 -0.14031369
## a3  0.04451039  0.02670623 -0.2017804 -0.06973294  0.22997033
## a4  0.18397626 -0.20389996 -0.1102162  0.01653243  0.04578211
## a5 -0.02670623 -0.10173802 -0.1360746  0.17041119 -0.06655362

det(A)      # Determinant of Matrix A

## [1] -4718

qr(A) $rank     # Matrix A Rank

## [1] 5

eigen(A)    # Eigenvalues and Eigenvectors of Matrix A

## eigen() decomposition
## $values
## [1] 20.689373+0.00000i -1.421612+5.02616i -1.421612-5.02616i -3.344927+0.00000i
## [5]  2.498778+0.00000i
## 
## $vectors
##              [,1]                  [,2]                  [,3]           [,4]
## [1,] 0.5470483+0i  0.4984887-0.1445515i  0.4984887+0.1445515i -0.52637480+0i
## [2,] 0.3359223+0i  0.0602646+0.1409489i  0.0602646-0.1409489i -0.27325187+0i
## [3,] 0.2240521+0i -0.1906021-0.4224541i -0.1906021+0.4224541i  0.70602430+0i
## [4,] 0.6684371+0i -0.0883510+0.3649339i -0.0883510-0.3649339i  0.38400538+0i
## [5,] 0.3014745+0i -0.5927362+0.0000000i -0.5927362+0.0000000i -0.04829652+0i
##                [,5]
## [1,] -0.32435065+0i
## [2,]  0.69826988+0i
## [3,] -0.05134298+0i
## [4,] -0.55000813+0i
## [5,] -0.31948518+0i

diag(A)     # Diagonal of Matrix A

## [1] 3 7 1 4 2

sum(diag(A))    # Trace of Matrix A

## [1] 17

rowSums(A)  # Sum of rows of Matrix A

## [1] 24 18 10 35 17

colSums(A)  # Sum of columns of Matrix A

## a1 a2 a3 a4 a5 
## 24 30 20 17 13

rowMeans(A) # Mean of the rows of Matrix A

## [1] 4.8 3.6 2.0 7.0 3.4

colMeans(A) # Mean of the columns of Matrix A

##  a1  a2  a3  a4  a5 
## 4.8 6.0 4.0 3.4 2.6

diag(1,3,3) # Identity Matrix  3x3

##      [,1] [,2] [,3]
## [1,]    1    0    0
## [2,]    0    1    0
## [3,]    0    0    1

A%*%B           # Matrix Product

##      [,1] [,2] [,3] [,4] [,5]
## [1,]  101  133   59   76  100
## [2,]   54  106   49   60   66
## [3,]   14   40   12   20   29
## [4,]  105  161  104  120  149
## [5,]   27   85   38   54   61

rbind(A,B)  # concatenate by rows of matrices

##    a1 a2 a3 a4 a5
##     3  8  3  9  1
##     2  7  4  3  2
##     5  3  1  1  0
##     8  9  6  4  8
##     6  3  6  0  2
## b1  0  1  0  1  3
## b2  1  7  2  2  1
## b3  2  9  3  5  4
## b4  9  5  3  4  7
## b5  6  2  7  6  8

cbind(A,B)  # concatenate by columns of matrices

##    a1 a2 a3 a4 a5          
## b1  3  8  3  9  1 0 1 0 1 3
## b2  2  7  4  3  2 1 7 2 2 1
## b3  5  3  1  1  0 2 9 3 5 4
## b4  8  9  6  4  8 9 5 3 4 7
## b5  6  3  6  0  2 6 2 7 6 8