I am a VBA user who started studying machine learning. As a memorandum, I would like to summarize the Python / R grammar while comparing it with VBA.
table of contents
-Matrix -Create Matrix -mxn matrix -Identity matrix -Zero Matrix -Diagonal matrix -Other -Matrix Attributes -Get Matrix Elements -Calculation of matrix -Calculation of Linear Algebra -Calculation for each element -Summary -List -Whole program
This time, I will talk about matrix calculation, which is indispensable for machine learning.
A = \begin{pmatrix}
1 & 2 & 3 & 4 \\
5 & 6 & 7 & 8 \\
9 & 10 & 11 & 12
\end{pmatrix}
Python
In Python, we use the numerical calculation library NumPy (import numpy). Numpy is often abbreviated as np (import numpy as np). Use a matrix as a two-dimensional array of Numpy.
Python3
import numpy as np
A = np.array(range(1,12+1))
print(A)
# [ 1 2 3 4 5 6 7 8 9 10 11 12]
A = np.array(range(1,12+1)).reshape(3,4)
A = np.array(range(1,12+1)).reshape(3,-1)
A = np.array(range(1,12+1)).reshape(-1,4)
print(A)
# [[ 1 2 3 4]
# [ 5 6 7 8]
# [ 9 10 11 12]]
A = np.array([[1,2,3,4],
[5,6,7,8],
[9,10,11,12]])
A = np.array([np.array(range(1,4+1)),
np.array(range(5,8+1)),
np.array(range(9,12+1))])
A = np.array([np.array(range(1,4+1)),
np.array(range(1,4+1))+4,
np.array(range(1,4+1))+8])
print(A)
# [[ 1 2 3 4]
# [ 5 6 7 8]
# [ 9 10 11 12]]
R
R
A <- matrix(c(1,2,3,4,5,6,7,8,9,10,11,12), 3, 4, byrow=TRUE)
A <- matrix(1:12, nrow=3, ncol=4, byrow=TRUE)
A <- matrix(1:12, nr=3, nc=4, b=T)
A <- matrix(1:12, 3, 4, byrow=TRUE)
A
# [,1] [,2] [,3] [,4]
# [1,] 1 2 3 4
# [2,] 5 6 7 8
# [3,] 9 10 11 12
A <- rbind(c(1,2,3,4),c(5,6,7,8),c(9,10,11,12))
A <- rbind(1:4,5:8,9:12)
A <- rbind(1:4,1:4+4,1:4+8)
A
# [,1] [,2] [,3] [,4]
# [1,] 1 2 3 4
# [2,] 5 6 7 8
# [3,] 9 10 11 12
A <- cbind(c(1,5,9),c(2,6,10),c(3,7,11),c(4,8,12))
A <- cbind(c(1,5,9),c(1,5,9)+1,c(1,5,9)+2,c(1,5,9)+3)
A
# [,1] [,2] [,3] [,4]
# [1,] 1 2 3 4
# [2,] 5 6 7 8
# [3,] 9 10 11 12
VBA
In VBA, for example, when the matrix A is entered in the B3: E5 cell range of an Excel worksheet, you can populate the array A as follows. In this case, the array A is two-dimensional and the indexes are 1 to 3 and 1 to 4.
VBA
Dim A As Variant
A = Range("B3:E5").Value
I_3 = \begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{pmatrix}
Python
Python3
print(np.eye(3))
# [[1. 0. 0.]
# [0. 1. 0.]
# [0. 0. 1.]]
eye is probably the identity matrix $ I $.
R
R
diag(3)
diag(1, 3)
diag(1, 3, 3)
# [,1] [,2] [,3]
# [1,] 1 0 0
# [2,] 0 1 0
# [3,] 0 0 1
VBA
In VBA, you can use Munit of WorksheetFunction (Excel worksheet function).
VBA
Dim Eye3 As Variant
Eye3 = WorksheetFunction.Munit(3)
O = \begin{pmatrix}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{pmatrix}
Python
Python3
print(np.zeros((3,4)))
# [[0. 0. 0. 0.]
# [0. 0. 0. 0.]
# [0. 0. 0. 0.]]
R
R
matrix(0, 3, 4)
diag(0, 3, 4)
# [,1] [,2] [,3] [,4]
# [1,] 0 0 0 0
# [2,] 0 0 0 0
# [3,] 0 0 0 0
diag(0, 3)
# [,1] [,2] [,3]
# [1,] 0 0 0
# [2,] 0 0 0
# [3,] 0 0 0
VBA
VBA
D = \begin{pmatrix}
1 & 0 & 0 \\
0 & 2 & 0 \\
0 & 0 & 3
\end{pmatrix}
Python
Python3
print(np.diag([1,2,3]))
# [[1 0 0]
# [0 2 0]
# [0 0 3]]
R
R
diag(c(1,2,3))
# [,1] [,2] [,3]
# [1,] 1 0 0
# [2,] 0 2 0
# [3,] 0 0 3
VBA
VBA
Python In Python, there are also functions that create such matrices.
Python3
print(np.ones((3,4)))
# [[1. 1. 1. 1.]
# [1. 1. 1. 1.]
# [1. 1. 1. 1.]]
print(np.full((3,4), 5))
# [[5 5 5 5]
# [5 5 5 5]
# [5 5 5 5]]
A = \begin{pmatrix}
1 & 2 & 3 & 4 \\
5 & 6 & 7 & 8 \\
9 & 10 & 11 & 12
\end{pmatrix}
Python
Python3
A = np.array(range(1,12+1)).reshape(3,4)
print(A)
# [[ 1 2 3 4]
# [ 5 6 7 8]
# [ 9 10 11 12]]
print(A.shape) # (Number of lines,Number of columns)
# (3, 4)
print(len(A))
print(A.shape[0]) #Number of lines
# 3
print(A.shape[1]) #Number of columns
# 4
print(A.ndim) #Dimension (matrix is two-dimensional)
# 2
print(type(A))
# <class 'numpy.ndarray'>
print(A.dtype)
# int32
R
R
A <- matrix(1:12, 3, 4, byrow=TRUE)
A
# [,1] [,2] [,3] [,4]
# [1,] 1 2 3 4
# [2,] 5 6 7 8
# [3,] 9 10 11 12
nrow(A) #Number of lines
# 3
ncol(A) #Number of columns
# 4
dim(A) #Number of lines,Number of columns
# 3 4
class(A)
# "matrix"
typeof(A)
# "integer"
VBA
For example, suppose matrix A is entered in the B3: E5 cell range of an Excel worksheet.
The array A is two-dimensional and has indexes 1 to 3 and 1 to 4.
VBA
Dim A As Variant
A = Range("B3:E5").Value
'Number of rows in the matrix
Debug.Print LBound(A, 1), UBound(A, 1), UBound(A, 1) - LBound(A, 1) + 1
'Number of columns in the matrix
Debug.Print LBound(A, 2), UBound(A, 2), UBound(A, 2) - LBound(A, 2) + 1
A = \begin{pmatrix}
1 & 2 & 3 & 4 \\
5 & 6 & 7 & 8 \\
9 & 10 & 11 & 12
\end{pmatrix}
Python
Python3
A = np.array(range(1,12+1)).reshape(3,4)
print(A)
# [[ 1 2 3 4]
# [ 5 6 7 8]
# [ 9 10 11 12]]
#component
print(A[0,1])
print(A[1-1,2-1])
print(A[0][1])
# 2
#Row vector
print(A[0,:])
# [1 2 3 4]
#Column vector
print(A[:,1])
# [ 2 6 10]
#Partially taken out
print(A[[0],:])
# [[1 2 3 4]]
print(A[:,[1]])
# [[ 2]
# [ 6]
# [10]]
print(A[0:2,])
# [[1 2 3 4]
# [5 6 7 8]]
print(A[:,1:4])
# [[ 2 3 4]
# [ 6 7 8]
# [10 11 12]]
print(A[-2:,:])
# [[ 5 6 7 8]
# [ 9 10 11 12]]
print(A[:,-2:])
# [[ 3 4]
# [ 7 8]
# [11 12]]
print(A[1:3, 2:4])
# [[ 7 8]
# [11 12]]
print(A[[0,2],:])
# [[ 1 2 3 4]
# [ 9 10 11 12]]
print(A[:,[1,3]])
# [[ 2 4]
# [ 6 8]
# [10 12]]
print(A[[0,2],[1,3]]) #This doesn't work
# [ 2 12]
print(A[np.ix_([0,2],[1,3])])
# [[ 2 4]
# [10 12]]
print(np.ix_([0,2],[1,3]))
# (array([[0],
# [2]]), array([[1, 3]]))
print(A[:,:])
# [[ 1 2 3 4]
# [ 5 6 7 8]
# [ 9 10 11 12]]
R
R
A <- matrix(1:12, 3, 4, byrow=TRUE)
A
# [,1] [,2] [,3] [,4]
# [1,] 1 2 3 4
# [2,] 5 6 7 8
# [3,] 9 10 11 12
#component
A[1,2]
# [1] 2
#Row vector
A[1,]
# [1] 1 2 3 4
#Column vector
A[,2]
# [1] 2 6 10
#Partially taken out
A[1:2,]
# [,1] [,2] [,3] [,4]
# [1,] 1 2 3 4
# [2,] 5 6 7 8
A[,2:3]
# [,1] [,2]
# [1,] 2 3
# [2,] 6 7
# [3,] 10 11
A[2:3,3:4]
# [,1] [,2]
# [1,] 7 8
# [2,] 11 12
A[c(1,3),]
#
# [1,] 1 2 3 4
# [2,] 9 10 11 12
A[,c(2,4)]
# [,1] [,2]
# [1,] 2 4
# [2,] 6 8
# [3,] 10 12
A[c(1,3),c(2,4)]
# [,1] [,2]
# [1,] 2 4
# [2,] 10 12
A[,]
# [,1] [,2] [,3] [,4]
# [1,] 1 2 3 4
# [2,] 5 6 7 8
# [3,] 9 10 11 12
head(A)
# [,1] [,2] [,3] [,4]
# [1,] 1 2 3 4
# [2,] 5 6 7 8
# [3,] 9 10 11 12
head(A, 2)
# [,1] [,2] [,3] [,4]
# [1,] 1 2 3 4
# [2,] 5 6 7 8
tail(A, 2)
# [,1] [,2] [,3] [,4]
# [2,] 5 6 7 8
# [3,] 9 10 11 12
VBA
For example, suppose matrix A is entered in the B3: E5 cell range of an Excel worksheet.
VBA
Dim A As Variant
A = Range("B3:E5").Value
'Of the procession(1, 2)component
Debug.Print A(1, 2)
A = \begin{pmatrix}
1 & 2 \\
3 & 4
\end{pmatrix}
,
B = \begin{pmatrix}
5 & 6 \\
7 & 8
\end{pmatrix}
k = 2,
kA = \begin{pmatrix}
2 & 4 \\
6 & 8
\end{pmatrix}
A + B = \begin{pmatrix}
6 & 8 \\
10 & 12
\end{pmatrix}
,
AB = \begin{pmatrix}
19 & 22 \\
43 & 50
\end{pmatrix}
A^T = \begin{pmatrix}
1 & 3 \\
2 & 4
\end{pmatrix}
,
A^{-1} = \begin{pmatrix}
-2 & 1 \\
1.5 & -0.5
\end{pmatrix}
detA = -2
,
trA = 5
Python
Python3
A = np.array(range(1,4+1)).reshape(2,2)
print(A)
# [[1 2]
# [3 4]]
B = np.array(range(5,8+1)).reshape(2,2)
print(B)
# [[5 6]
# [7 8]]
#Matrix scalar times
k = 2
print(A * k)
# [[2 4]
# [6 8]]
#Matrix sum
print(A + B)
# [[ 6 8]
# [10 12]]
#Matrix multiplication
print(A @ B)
# [[19 22]
# [43 50]]
print(np.dot(A, B))
# [[19 22]
# [43 50]]
#Transpose matrix
print(A.transpose())
# [[1 3]
# [2 4]]
print(A.T)
# [[1 3]
# [2 4]]
#Inverse matrix
print(np.linalg.inv(A))
# [[-2. 1. ]
# [ 1.5 -0.5]]
#Identity matrix
print(np.eye(2))
# [[1. 0.]
# [0. 1.]]
#Determinant detA
print(np.linalg.det(A))
# -2.0000000000000004
#Diagonal component
print(np.diag(A))
# [1 4]
#Trace (sum of diagonal components) trA
print(sum(np.diag(A)))
# 5
#Matrix flattening (vectorization)
print(A.flatten())
# [1 2 3 4]
#Matrix transformation
print(A.reshape(1,4))
# [[1 2 3 4]]
print(A.reshape(4,1))
# [[1]
# [2]
# [3]
# [4]]
R
R
A <- matrix(1:4, 2, 2, TRUE)
A
# [,1] [,2]
# [1,] 1 2
# [2,] 3 4
B <- matrix(5:8, 2, 2, T)
B
# [,1] [,2]
# [1,] 5 6
# [2,] 7 8
#Matrix scalar times
k <- 2
A * k
# [,1] [,2]
# [1,] 2 4
# [2,] 6 8
#Matrix sum
A + B
# [,1] [,2]
# [1,] 6 8
# [2,] 10 12
#Matrix multiplication
A %*% B
# [,1] [,2]
# [1,] 19 22
# [2,] 43 50
#Transpose matrix
t(A)
# [,1] [,2]
# [1,] 1 3
# [2,] 2 4
#Inverse matrix
solve(A)
# [,1] [,2]
# [1,] -2.0 1.0
# [2,] 1.5 -0.5
#Identity matrix
diag(2)
# [,1] [,2]
# [1,] 1 0
# [2,] 0 1
#Determinant detA
det(A)
# -2
#Diagonal component
diag(A)
# [1] 1 4
#Trace (sum of diagonal components) trA
sum(diag(A))
# 5
VBA
For example, suppose you have a matrix A in the B3: C5 cell range of an Excel worksheet and a matrix B in the B8: C9 cell range.
VBA
Dim A As Variant
Dim B As Variant
Dim k As Double
Dim kA As Variant
Dim ApB As Variant
Dim AB As Variant
Dim At As Variant
Dim Ainv As Variant
Dim Eye2 As Variant
Dim detA As Double
Dim trA As Double
Dim i As Integer
Dim j As Integer
'queue
A = Range("B3:E5").Value
B = Range("B8:C9").Value
'Number of rows in the matrix
Debug.Print LBound(A, 1), UBound(A, 1), UBound(A, 1) - LBound(A, 1) + 1
'Number of columns in the matrix
Debug.Print LBound(A, 2), UBound(A, 2), UBound(A, 2) - LBound(A, 2) + 1
'Matrix element A(1, 2)
Debug.Print A(1, 2)
'Matrix scalar times kA
k = 2
ReDim kA(LBound(A, 1) To UBound(A, 1), LBound(A, 2) To UBound(A, 2))
For i = LBound(A, 1) To UBound(A, 1)
For j = LBound(A, 2) To UBound(A, 2)
kA(i, j) = A(i, j) * k
Next j
Next i
Range("E13:F14").Value = kA
'Matrix sum A+B
ReDim ApB(LBound(A, 1) To UBound(A, 1), LBound(A, 2) To UBound(A, 2))
For i = LBound(A, 1) To UBound(A, 1)
For j = LBound(A, 2) To UBound(A, 2)
ApB(i, j) = A(i, j) + B(i, j)
Next j
Next i
Range("E18:F19").Value = ApB
'Matrix product AB
AB = WorksheetFunction.MMult(A, B)
Range("E18:F19").Value = AB
'Transpose matrix A^T
At = WorksheetFunction.Transpose(A)
Range("E28:F29").Value = At
'Inverse matrix A^(-1)
Ainv = WorksheetFunction.MInverse(A)
Range("E33:F34").Value = Ainv
'Identity matrix
Eye2 = WorksheetFunction.Munit(2)
Range("E38:F39").Value = Eye2
'Determinant detA
detA = WorksheetFunction.MDeterm(A)
Range("E42").Value = detA
'Trace trA
trA = WorksheetFunction.SumProduct(A, Eye2)
Range("E45").Value = trA
In Python and R, apart from the calculation of linear algebra, you can calculate each element in the same way as the scalar operation.
Python
Python3
print(A + B)
# [[ 6 8]
# [10 12]]
print(A - B)
# [[-4 -4]
# [-4 -4]]
print(A * B)
# [[ 5 12]
# [21 32]]
print(A / B)
# [[0.2 0.33333333]
# [0.42857143 0.5 ]]
print(A * 2)
# [[2 4]
# [6 8]]
print(A ** 2)
# [[ 1 4]
# [ 9 16]]
print(-A)
# [[-1 -2]
# [-3 -4]]
print(A == B)
# [[False False]
# [False False]]
print(A > 2)
# [[False False]
# [ True True]]
print(A % 2 == 0)
# [[False True]
# [False True]]
print(np.exp(A))
# [[ 2.71828183 7.3890561 ]
# [20.08553692 54.59815003]]
print(np.log(A))
# [[0. 0.69314718]
# [1.09861229 1.38629436]]
R
R
A + B
# [,1] [,2]
# [1,] 6 8
# [2,] 10 12
A - B
# [,1] [,2]
# [1,] -4 -4
# [2,] -4 -4
A * B
# [,1] [,2]
# [1,] 5 12
# [2,] 21 32
A / B
# [,1] [,2]
# [1,] 0.2000000 0.3333333
# [2,] 0.4285714 0.5000000
A * 2
# [,1] [,2]
# [1,] 2 4
# [2,] 6 8
A ^ 2
# [,1] [,2]
# [1,] 1 4
# [2,] 9 16
-A
# [,1] [,2]
# [1,] -1 -2
# [2,] -3 -4
A == B
# [,1] [,2]
# [1,] FALSE FALSE
# [2,] FALSE FALSE
A < B
# [,1] [,2]
# [1,] TRUE TRUE
# [2,] TRUE TRUE
A > 2
# [,1] [,2]
# [1,] FALSE FALSE
# [2,] TRUE TRUE
A %% 2 == 0
# [,1] [,2]
# [1,] FALSE TRUE
# [2,] FALSE TRUE
exp(A)
# [,1] [,2]
# [1,] 2.718282 7.389056
# [2,] 20.085537 54.598150
log(A)
# [,1] [,2]
# [1,] 0.000000 0.6931472
# [2,] 1.098612 1.3862944
VBA
In VBA, you only have to calculate for each (i, j) component in the for statement.
VBA
List the functions used in each language. For comparison, the calculation in EXCEL is also shown.
In Python, use Numpy (import numpy as np).
*** Creating a matrix ***
A = \begin{pmatrix}
1 & 2 & 3 & 4 \\
5 & 6 & 7 & 8 \\
9 & 10 & 11 & 12
\end{pmatrix}
,
I_3 = \begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{pmatrix}
O = \begin{pmatrix}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{pmatrix}
,
D = \begin{pmatrix}
1 & 0 & 0 \\
0 & 2 & 0 \\
0 & 0 & 3
\end{pmatrix}
| queue | Python | R | VBA | EXCEL |
|---|---|---|---|---|
| mxn matrix |
np.array(range(1,12+1)).reshape(3,4)np.array([[1,2,3,4],[5,6,7,8],[9,10,11,12]]) |
matrix(1:12,3,4,TRUE)rbind(c(1,2,3,4),c(5,6,7,8),c(9,10,11,12))cbind(c(1,5,9),c(2,6,10),c(3,7,11),c(4,8,12)) |
||
| Identity matrix |
np.eye(3) |
diag(3)diag(1, 3)diag(1, 3, 3) |
WorksheetFunction.Munit(3) |
MUNIT(3) |
| Zero matrix |
np.zeros((3,4)) |
matrix(0, 3, 4)diag(0, 3, 4)diag(0, 3) |
||
| Diagonal matrix |
np.diag([1,2,3] |
diag(c(1,2,3)) |
||
| 1 padded matrix | np.ones((3,4)) |
|||
| Numeric padded matrix | np.full((3,4), 5) |
*** Matrix attributes ***
A = \begin{pmatrix}
1 & 2 & 3 & 4 \\
5 & 6 & 7 & 8 \\
9 & 10 & 11 & 12
\end{pmatrix}
| Python | R | VBA | EXCEL | result | |
|---|---|---|---|---|---|
| form | A.shape |
dim(A) |
3, 4 | ||
| Number of lines | len(A)A.shape[0] |
nrow(A) |
UBound(A, 1) |
3 | |
| Number of columns | A.shape[1] |
ncol(A) |
UBound(A, 2) |
4 | |
| dimension(=2) | A.ndim |
2 | |||
| type | type(A) |
class(A) |
|||
| Data type | A.dtype |
typeof(A) |
*** Matrix elements ***
A = \begin{pmatrix}
1 & 2 & 3 & 4 \\
5 & 6 & 7 & 8 \\
9 & 10 & 11 & 12
\end{pmatrix}
| Python | R | VBA | EXCEL | |
|---|---|---|---|---|
| (1,2)component | A[0,1]A[0][1] |
A[1,2] |
A(1,2) |
=INDEX(A,1,2) |
| 1st row vector | A[0,:]A[[0],:] |
A[1,] |
||
| Second column vector | A[:,1]A[:,[1]] |
A[,2] |
||
| 1~2 lines | A[0:2,] |
A[1:2,] |
||
| 3~4 rows | A[:,2:4]A[:,-2:] |
A[,3:4] |
||
| 1~2 lines,3~4 rows | A[0:2,2:4]A[:2,-2:] |
A[1:2,3:4] |
||
| Lines 1 and 3 | A[[0,2],:] |
A[c(1,3),] |
||
| 2 and 4 rows | A[:,[1,3]] |
A[,c(2,4)] |
||
| Lines 1 and 3,2 and 4 rows | A[np.ix_([0,2],[1,3])] |
A[c(1,3),c(2,4)] |
||
| All lines,All rows | A[:,:] |
A[,] |
||
| head | head(A)head(A, 2) |
|||
| tail | tail(A)tail(A, 2) |
*** Linear algebra calculation ***
A = \begin{pmatrix}
1 & 2 \\
3 & 4
\end{pmatrix}
,
B = \begin{pmatrix}
5 & 6 \\
7 & 8
\end{pmatrix}
k = 2,
kA = \begin{pmatrix}
2 & 4 \\
6 & 8
\end{pmatrix}
A + B = \begin{pmatrix}
6 & 8 \\
10 & 12
\end{pmatrix}
,
AB = \begin{pmatrix}
19 & 22 \\
43 & 50
\end{pmatrix}
A^T = \begin{pmatrix}
1 & 3 \\
2 & 4
\end{pmatrix}
,
A^{-1} = \begin{pmatrix}
-2 & 1 \\
1.5 & -0.5
\end{pmatrix}
detA = -2
,
trA = 5
| Python | R | VBA | EXCEL | |
|---|---|---|---|---|
| Scalar times |
A * k |
A * k |
=cell*k |
|
| Matrix sum |
A + B |
A + B |
=cell+cell |
|
| Matrix multiplication |
A @ Bnp.dot(A, B) |
A %*% B |
WorksheetFunction.MMult(A, B) |
=MMULT(A,B) |
| Transpose matrix |
A.transpose()A.T |
t(A) |
WorksheetFunction.Transpose(A) |
=Transpose(A) |
| Inverse matrix |
np.linalg.inv(A) |
solve(A) |
WorksheetFunction.MInverse(A) |
=MINVERSE(A) |
| Identity matrix |
np.eye(2) |
diag(2) |
WorksheetFunction.Munit(2) |
=MUNIT(2) |
| Determinant |
np.linalg.det(A) |
det(A) |
WorksheetFunction.MDeterm(A) |
=MDETERM(A) |
| trace |
sum(np.diag(A)) |
sum(diag(A)) |
WorksheetFunction.SumProduct(A,Eye2) |
|
| Diagonal component | np.diag(A) |
diag(A) |
||
| Matrix flattening | A.flatten() |
|||
| Matrix transformation | A.reshape(1,4) |
*** Arithmetic for each element of the matrix ***
| Python | R | VBA | EXCEL | |
|---|---|---|---|---|
| sum | A + B |
A + B |
=cell+cell |
|
| difference | A - B |
A - B |
=cell-cell |
|
| product | A * B |
A * B |
=cell*cell |
|
| quotient | A / B |
A / B |
=cell/cell |
|
| Scalar times | A * 2 |
A * 2 |
=cell*2 |
|
| Exponentiation | A ** 2 |
A ^ 2 |
=cell^2 |
|
| Sign inversion | -A |
-A |
=-cell |
|
| equal sign | A == B |
A == B |
=cell=cell |
|
| Inequality sign | A < B |
A < B |
=cell<cell |
|
| Inequality sign | A > 2 |
A > 2 |
=cell>2 |
|
| Remainder 0 | A % 2 == 0 |
A %% 2 == 0 |
=MOD(cell,2)=0 |
|
| square root | np.sqrt(A) |
sqrt(A) |
=SQRT(cell) |
|
| exp | np.exp(A) |
exp(A) |
=EXP(cell) |
|
| log | np.log(A) |
log(A) |
=LN(cell) |
The whole program used for reference is shown.
Python
Python3
import numpy as np
#Creating a matrix
#mxn matrix
A = np.array(range(1,12+1))
print(A)
# [ 1 2 3 4 5 6 7 8 9 10 11 12]
A = np.array(range(1,12+1)).reshape(3,4)
A = np.array(range(1,12+1)).reshape(3,-1)
A = np.array(range(1,12+1)).reshape(-1,4)
print(A)
# [[ 1 2 3 4]
# [ 5 6 7 8]
# [ 9 10 11 12]]
A = np.array([[1,2,3,4],
[5,6,7,8],
[9,10,11,12]])
A = np.array([np.array(range(1,4+1)),
np.array(range(5,8+1)),
np.array(range(9,12+1))])
A = np.array([np.array(range(1,4+1)),
np.array(range(1,4+1))+4,
np.array(range(1,4+1))+8])
print(A)
# [[ 1 2 3 4]
# [ 5 6 7 8]
# [ 9 10 11 12]]
#Identity matrix
print(np.eye(3))
# [[1. 0. 0.]
# [0. 1. 0.]
# [0. 0. 1.]]
#Zero matrix
print(np.zeros((3,4)))
# [[0. 0. 0. 0.]
# [0. 0. 0. 0.]
# [0. 0. 0. 0.]]
print(np.zeros((3,3)))
# [[0. 0. 0.]
# [0. 0. 0.]
# [0. 0. 0.]]
#Diagonal matrix
print(np.diag([1,2,3]))
# [[1 0 0]
# [0 2 0]
# [0 0 3]]
#Other
print(np.ones((3,4)))
# [[1. 1. 1. 1.]
# [1. 1. 1. 1.]
# [1. 1. 1. 1.]]
print(np.full((3,4), 5))
# [[5 5 5 5]
# [5 5 5 5]
# [5 5 5 5]]
#Matrix attributes
A = np.array(range(1,12+1)).reshape(3,4)
print(A)
# [[ 1 2 3 4]
# [ 5 6 7 8]
# [ 9 10 11 12]]
print(A.shape) # (Number of lines,Number of columns)
# (3, 4)
print(len(A))
print(A.shape[0]) #Number of lines
# 3
print(A.shape[1]) #Number of columns
# 4
print(A.ndim) #Dimension (matrix is two-dimensional)
# 2
print(type(A))
# <class 'numpy.ndarray'>
print(A.dtype)
# int32
#Get matrix elements
A = np.array(range(1,12+1)).reshape(3,4)
print(A)
# [[ 1 2 3 4]
# [ 5 6 7 8]
# [ 9 10 11 12]]
#component
print(A[0,1])
print(A[1-1,2-1])
print(A[0][1])
# 2
#Row vector
print(A[0,:])
# [1 2 3 4]
#Column vector
print(A[:,1])
# [ 2 6 10]
print(A[[0],:])
# [[1 2 3 4]]
print(A[:,[1]])
# [[ 2]
# [ 6]
# [10]]
print(A[0:2,])
print(A[:2,])
# [[1 2 3 4]
# [5 6 7 8]]
print(A[:,2:4])
print(A[:,-2:])
# [[ 3 4]
# [ 7 8]
# [11 12]]
print(A[0:2,2:4])
print(A[:2,-2:])
# [[3 4]
# [7 8]]
print(A[[0,2],:])
# [[ 1 2 3 4]
# [ 9 10 11 12]]
print(A[:,[1,3]])
# [[ 2 4]
# [ 6 8]
# [10 12]]
print(A[[0,2],[1,3]]) #This doesn't work
# [ 2 12]
print(A[np.ix_([0,2],[1,3])])
# [[ 2 4]
# [10 12]]
print(np.ix_([0,2],[1,3]))
# (array([[0],
# [2]]), array([[1, 3]]))
print(A[:,:])
# [[ 1 2 3 4]
# [ 5 6 7 8]
# [ 9 10 11 12]]
#Calculation of linear algebra
A = np.array(range(1,4+1)).reshape(2,2)
print(A)
# [[1 2]
# [3 4]]
B = np.array(range(5,8+1)).reshape(2,2)
print(B)
# [[5 6]
# [7 8]]
#Matrix scalar times
k = 2
print(A * k)
# [[2 4]
# [6 8]]
#Matrix sum
print(A + B)
# [[ 6 8]
# [10 12]]
#Matrix multiplication
print(A @ B)
# [[19 22]
# [43 50]]
print(np.dot(A, B))
# [[19 22]
# [43 50]]
#Transpose matrix
print(A.transpose())
# [[1 3]
# [2 4]]
print(A.T)
# [[1 3]
# [2 4]]
#Inverse matrix
print(np.linalg.inv(A))
# [[-2. 1. ]
# [ 1.5 -0.5]]
#Identity matrix
print(np.eye(2))
# [[1. 0.]
# [0. 1.]]
#Determinant detA
print(np.linalg.det(A))
# -2.0000000000000004
#Diagonal component
print(np.diag(A))
# [1 4]
#Trace (sum of diagonal components) trA
print(sum(np.diag(A)))
# 5
#Matrix flattening (vectorization)
print(A.flatten())
# [1 2 3 4]
#Matrix transformation
print(A.reshape(1,4))
# [[1 2 3 4]]
print(A.reshape(4,1))
# [[1]
# [2]
# [3]
# [4]]
#Operations for each element of the matrix
print(A + B)
# [[ 6 8]
# [10 12]]
print(A - B)
# [[-4 -4]
# [-4 -4]]
print(A * B)
# [[ 5 12]
# [21 32]]
print(A / B)
# [[0.2 0.33333333]
# [0.42857143 0.5 ]]
print(A * 2)
# [[2 4]
# [6 8]]
print(A ** 2)
# [[ 1 4]
# [ 9 16]]
print(-A)
# [[-1 -2]
# [-3 -4]]
print(A == B)
# [[False False]
# [False False]]
print(A < B)
# [[ True True]
# [ True True]]
print(A > 2)
# [[False False]
# [ True True]]
print(A % 2 == 0)
# [[False True]
# [False True]]
print(np.sqrt(A))
# [[1. 1.41421356]
# [1.73205081 2. ]]
print(np.exp(A))
# [[ 2.71828183 7.3890561 ]
# [20.08553692 54.59815003]]
print(np.log(A))
# [[0. 0.69314718]
# [1.09861229 1.38629436]]
R
R
#Creating a matrix
#mxn matrix
A <- matrix(c(1,2,3,4,5,6,7,8,9,10,11,12), 3, 4, byrow=TRUE)
A <- matrix(1:12, nrow=3, ncol=4, byrow=TRUE)
A <- matrix(1:12, nr=3, nc=4, b=T)
A <- matrix(1:12, 3, 4, byrow=TRUE)
A
# [,1] [,2] [,3] [,4]
# [1,] 1 2 3 4
# [2,] 5 6 7 8
# [3,] 9 10 11 12
A <- rbind(c(1,2,3,4),c(5,6,7,8),c(9,10,11,12))
A <- rbind(1:4,5:8,9:12)
A <- rbind(1:4,1:4+4,1:4+8)
A
# [,1] [,2] [,3] [,4]
# [1,] 1 2 3 4
# [2,] 5 6 7 8
# [3,] 9 10 11 12
A <- cbind(c(1,5,9),c(2,6,10),c(3,7,11),c(4,8,12))
A <- cbind(c(1,5,9),c(1,5,9)+1,c(1,5,9)+2,c(1,5,9)+3)
A
# [,1] [,2] [,3] [,4]
# [1,] 1 2 3 4
# [2,] 5 6 7 8
# [3,] 9 10 11 12
#Identity matrix
diag(3)
diag(1, 3)
diag(1, 3, 3)
# [,1] [,2] [,3]
# [1,] 1 0 0
# [2,] 0 1 0
# [3,] 0 0 1
#Zero matrix
matrix(0, 3, 4)
diag(0, 3, 4)
# [,1] [,2] [,3] [,4]
# [1,] 0 0 0 0
# [2,] 0 0 0 0
# [3,] 0 0 0 0
diag(0, 3)
# [,1] [,2] [,3]
# [1,] 0 0 0
# [2,] 0 0 0
# [3,] 0 0 0
#Diagonal matrix
diag(c(1,2,3))
# [,1] [,2] [,3]
# [1,] 1 0 0
# [2,] 0 2 0
# [3,] 0 0 3
#Matrix attributes
A <- matrix(1:12, 3, 4, byrow=TRUE)
A
# [,1] [,2] [,3] [,4]
# [1,] 1 2 3 4
# [2,] 5 6 7 8
# [3,] 9 10 11 12
nrow(A) #Number of lines
# 3
ncol(A) #Number of columns
# 4
dim(A) #Number of lines,Number of columns
# 3 4
class(A)
# "matrix"
typeof(A)
# "integer"
#Get matrix elements
A <- matrix(1:12, 3, 4, byrow=TRUE)
A
# [,1] [,2] [,3] [,4]
# [1,] 1 2 3 4
# [2,] 5 6 7 8
# [3,] 9 10 11 12
A[1,2]
# [1] 2
#Row vector
A[1,]
# [1] 1 2 3 4
#Column vector
A[,2]
# [1] 2 6 10
#part
A[1:2,]
# [,1] [,2] [,3] [,4]
# [1,] 1 2 3 4
# [2,] 5 6 7 8
A[,3:4]
# [,1] [,2]
# [1,] 3 4
# [2,] 7 8
# [3,] 11 12
A[1:2,3:4]
# [,1] [,2]
# [1,] 3 4
# [2,] 7 8
A[c(1,3),]
#
# [1,] 1 2 3 4
# [2,] 9 10 11 12
A[,c(2,4)]
# [,1] [,2]
# [1,] 2 4
# [2,] 6 8
# [3,] 10 12
A[c(1,3),c(2,4)]
# [,1] [,2]
# [1,] 2 4
# [2,] 10 12
A[,]
# [,1] [,2] [,3] [,4]
# [1,] 1 2 3 4
# [2,] 5 6 7 8
# [3,] 9 10 11 12
head(A)
# [,1] [,2] [,3] [,4]
# [1,] 1 2 3 4
# [2,] 5 6 7 8
# [3,] 9 10 11 12
head(A, 2)
# [,1] [,2] [,3] [,4]
# [1,] 1 2 3 4
# [2,] 5 6 7 8
tail(A, 2)
# [,1] [,2] [,3] [,4]
# [2,] 5 6 7 8
# [3,] 9 10 11 12
#Calculation of linear algebra
A <- matrix(1:4, 2, 2, TRUE)
A
# [,1] [,2]
# [1,] 1 2
# [2,] 3 4
B <- matrix(5:8, 2, 2, T)
B
# [,1] [,2]
# [1,] 5 6
# [2,] 7 8
#Matrix scalar times
k <- 2
A * k
# [,1] [,2]
# [1,] 2 4
# [2,] 6 8
#Matrix sum
A + B
# [,1] [,2]
# [1,] 6 8
# [2,] 10 12
#Matrix multiplication
A %*% B
# [,1] [,2]
# [1,] 19 22
# [2,] 43 50
#Transpose matrix
t(A)
# [,1] [,2]
# [1,] 1 3
# [2,] 2 4
#Inverse matrix
solve(A)
# [,1] [,2]
# [1,] -2.0 1.0
# [2,] 1.5 -0.5
#Identity matrix
diag(2)
# [,1] [,2]
# [1,] 1 0
# [2,] 0 1
#Determinant detA
det(A)
# -2
#Diagonal component
diag(A)
# [1] 1 4
#Trace (sum of diagonal components) trA
sum(diag(A))
# 5
#Operations for each element of the matrix
A + B
# [,1] [,2]
# [1,] 6 8
# [2,] 10 12
A - B
# [,1] [,2]
# [1,] -4 -4
# [2,] -4 -4
A * B
# [,1] [,2]
# [1,] 5 12
# [2,] 21 32
A / B
# [,1] [,2]
# [1,] 0.2000000 0.3333333
# [2,] 0.4285714 0.5000000
A * 2
# [,1] [,2]
# [1,] 2 4
# [2,] 6 8
A ^ 2
# [,1] [,2]
# [1,] 1 4
# [2,] 9 16
-A
# [,1] [,2]
# [1,] -1 -2
# [2,] -3 -4
A == B
# [,1] [,2]
# [1,] FALSE FALSE
# [2,] FALSE FALSE
A < B
# [,1] [,2]
# [1,] TRUE TRUE
# [2,] TRUE TRUE
A > 2
# [,1] [,2]
# [1,] FALSE FALSE
# [2,] TRUE TRUE
A %% 2 == 0
# [,1] [,2]
# [1,] FALSE TRUE
# [2,] FALSE TRUE
sqrt(A)
# [,1] [,2]
# [1,] 1.000000 1.414214
# [2,] 1.732051 2.000000
exp(A)
# [,1] [,2]
# [1,] 2.718282 7.389056
# [2,] 20.085537 54.598150
log(A)
# [,1] [,2]
# [1,] 0.000000 0.6931472
# [2,] 1.098612 1.3862944
VBA
VBA
Sub test_matrix()
Dim A As Variant
Dim B As Variant
Dim k As Double
Dim kA As Variant
Dim ApB As Variant
Dim AB As Variant
Dim At As Variant
Dim Ainv As Variant
Dim Eye2 As Variant
Dim detA As Double
Dim trA As Double
Dim i As Integer
Dim j As Integer
'queue
A = Range("B3:E5").Value
print_matrix (A)
B = Range("B8:C9").Value
print_matrix (B)
'Number of rows in the matrix
Debug.Print LBound(A, 1), UBound(A, 1), UBound(A, 1) - LBound(A, 1) + 1
'Number of columns in the matrix
Debug.Print LBound(A, 2), UBound(A, 2), UBound(A, 2) - LBound(A, 2) + 1
'Matrix element A(1, 2)
Debug.Print A(1, 2)
'Matrix scalar times kA
k = 2
ReDim kA(LBound(A, 1) To UBound(A, 1), LBound(A, 2) To UBound(A, 2))
For i = LBound(A, 1) To UBound(A, 1)
For j = LBound(A, 2) To UBound(A, 2)
kA(i, j) = A(i, j) * k
Next j
Next i
Call print_matrix(kA)
Range("E13:F14").Value = kA
'Matrix sum A+B
ReDim ApB(LBound(A, 1) To UBound(A, 1), LBound(A, 2) To UBound(A, 2))
For i = LBound(A, 1) To UBound(A, 1)
For j = LBound(A, 2) To UBound(A, 2)
ApB(i, j) = A(i, j) + B(i, j)
Next j
Next i
Call print_matrix(ApB)
Range("E18:F19").Value = ApB
'Matrix product AB
AB = WorksheetFunction.MMult(A, B)
Call print_matrix(AB)
Range("E18:F19").Value = AB
'Transpose matrix A^T
At = WorksheetFunction.Transpose(A)
Call print_matrix(At)
Range("E28:F29").Value = At
'Inverse matrix A^(-1)
Ainv = WorksheetFunction.MInverse(A)
Call print_matrix(Ainv)
Range("E33:F34").Value = Ainv
'Identity matrix
Eye2 = WorksheetFunction.Munit(2)
Call print_matrix(Eye2)
Range("E38:F39").Value = Eye2
'Determinant detA
detA = WorksheetFunction.MDeterm(A)
Debug.Print detA
Range("E42").Value = detA
'Trace trA
trA = WorksheetFunction.SumProduct(A, Eye2)
Debug.Print trA
Range("E45").Value = trA
End Sub
Sub print_matrix(M As Variant)
'For matrix display
Dim i As Integer
Dim j As Integer
Dim s As String
If IsArray(M) Then
s = ""
For i = LBound(M, 1) To UBound(M, 1)
For j = LBound(M, 2) To UBound(M, 2)
s = s & vbTab & M(i, j)
Next j
s = s & vbCrLf
Next i
Debug.Print s
End If
End Sub
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