Vectors and Matrices — SepalSolver

Vectors and Matrices are fundamental to Linear Algebra. SepalSolver provides three array types: RowVec, ColVec and Matrix. RowVec and ColVec are 1D arrays while Matrix is a 2D array.

Creating Vectors and Matrices

Example 1C#

// Row vector
RowVec R = new double[] { 5, 6, 7, 1 };
Console.WriteLine($"R = {R}");
// Column vector
ColVec C = new double[] { 8, 3, 4, 2, 7 };
Console.WriteLine($"C = {C}");
// Matrix
Matrix M = new double[,] 
{
    {5, -2, 3, 7 },
    {2, 1, -7, 3 },
    {4, 8, 9, 1 },
    {0, 5, -6, -3 }
};
Console.WriteLine($"M = {M}");

OutputFrom the book

R = 
 5   6   7   1 

C = 
 8 
 3 
 4 
 2 
 7 

M = 
 5  -2   3   7 
 2   1  -7   3 
 4   8   9   1 
 0   5  -6  -3

Vectors and Matrices can also be initialized using random

Example 2C#

// Row vector
RowVec R = Rand(7);
Console.WriteLine($"R = {R}");
// Column vector
ColVec C = Rand(5);
Console.WriteLine($"C = {C}");
// Matrix
Matrix M = Rand(8, 7);
Console.WriteLine($"M = {M}");

OutputFrom the book

R = 
   0.3408    0.4771    0.2018    0.1315    0.8247    0.7431    0.0112

C = 
   0.6404
   0.3563
   0.4054
   0.6549
   0.5426

M = 
   0.2200    0.3615    0.6376    0.1093    0.4242    0.5938    0.3550
   0.2720    0.9079    0.7637    0.3128    0.9002    0.8761    0.1103
   0.8454    0.1511    0.5838    0.4349    0.8613    0.2663    0.4054
   0.3745    0.3098    0.9405    0.3829    0.0364    0.0178    0.9488
   0.3864    0.4172    0.1403    0.4578    0.0883    0.2745    0.1012
   0.9598    0.7470    0.3624    0.5528    0.8384    0.5277    0.8321
   0.0870    0.4344    0.9954    0.4480    0.1671    0.4440    0.2958
   0.2129    0.9812    0.8039    0.6421    0.5354    0.7338    0.5285

Vectors can be initialized using Zeros, Ones, Eye etc

Example 3C#

// Row vector
RowVec R = Zeros(7);
Console.WriteLine($"R = {R}");
// Column vector
ColVec C = Ones(5);
Console.WriteLine($"C = {C}");
// Matrix
Matrix M = Eye(7, 7);
Console.WriteLine($"M = {M}");

OutputFrom the book

R = 
 0   0   0   0   0   0   0 

C = 
 1 
 1 
 1 
 1 
 1 

M = 
 1   0   0   0   0   0   0 
 0   1   0   0   0   0   0 
 0   0   1   0   0   0   0 
 0   0   0   1   0   0   0 
 0   0   0   0   1   0   0 
 0   0   0   0   0   1   0 
 0   0   0   0   0   0   1

Vectors and Matrices can be concatenated

Example 4C#

RowVec R1 = Rand(4);
Console.WriteLine($"R1 = {R1}");
RowVec R2 = Rand(5);
Console.WriteLine($"R2 = {R2}");
// Horizontal concatenation
RowVec R3 = Hcart(R1, R2);
Console.WriteLine($"R3 = {R3}");
ColVec C1 = Rand(10);
Console.WriteLine($"C1 = {C1}");
ColVec C2 = Rand(10);
Console.WriteLine($"C2 = {C2}");
// Horizontal concatenation
Matrix M = Hcart(C1, C2);
Console.WriteLine($"M = {M}");

OutputFrom the book

R1 = 
   0.9723    0.5128    0.2176    0.5148

R2 = 
   0.3392    0.5221    0.9440    0.2666    0.8323

R3 = 
   0.9723    0.5128    0.2176    0.5148    0.3392    0.5221    0.9440    0.2666    0.8323

C1 = 
   0.0111
   0.9927
   0.1004
   0.0450
   0.6581
   0.4342
   0.9595
   0.2445
   0.5650
   0.5547

C2 = 
   0.2701
   0.8769
   0.4659
   0.5733
   0.4500
   0.7537
   0.0837
   0.9517
   0.2559
   0.8796

M = 
   0.0111    0.2701
   0.9927    0.8769
   0.1004    0.4659
   0.0450    0.5733
   0.6581    0.4500
   0.4342    0.7537
   0.9595    0.0837
   0.2445    0.9517
   0.5650    0.2559
   0.5547    0.8796

Vertical Concatenation

Example 5C#

RowVec R1 = Rand(4);
Console.WriteLine($"R1 = {R1}");
RowVec R2 = Rand(4);
Console.WriteLine($"R2 = {R2}");
// Vertical concatenation
Matrix M = Vcart(R1, R2);
Console.WriteLine($"M = {M}");
ColVec C1 = Rand(10);
Console.WriteLine($"C1 = {C1}");
ColVec C2 = Rand(2);
Console.WriteLine($"C2 = {C2}");
// Vertical concatenation
ColVec C3 = Vcart(C1, C2);
Console.WriteLine($"C3 = {C3}");

OutputFrom the book

R1 = 
   0.3412    0.2229    0.7419    0.5755

R2 = 
   0.4963    0.0361    0.2773    0.9844

M = 
   0.3412    0.2229    0.7419    0.5755
   0.4963    0.0361    0.2773    0.9844

C1 = 
   0.6298
   0.1612
   0.8009
   0.7666
   0.6063
   0.1633
   0.7198
   0.5983
   0.0308
   0.4191

C2 = 
   0.4352
   0.6507

C3 = 
   0.6298
   0.1612
   0.8009
   0.7666
   0.6063
   0.1633
   0.7198
   0.5983
   0.0308
   0.4191
   0.4352
   0.6507

Flipping a Matrix

We can flip a Matrix vertically (flipud) or horizontally (fliplr).

Example 6C#

Matrix M = new double[,]
{
    {5, -2, 3, 7 },
    {2, 1, -7, 3 },
    {4, 8, 9, 1 },
    {0, 5, -6, -3 }
};
Console.WriteLine($"M = {M}");
Console.WriteLine($"Flipud(M) = {Flipud(M)}");
Console.WriteLine($"Fliplr(M) = {Fliplr(M)}");

OutputFrom the book

M = 
 5  -2   3   7 
 2   1  -7   3 
 4   8   9   1 
 0   5  -6  -3 

Flipud(M) = 
 0   5  -6  -3 
 4   8   9   1 
 2   1  -7   3 
 5  -2   3   7 

Fliplr(M) = 
 7   3  -2   5 
 3  -7   1   2 
 1   9   8   4 
-3  -6   5   0

Extract a Triangular Portion of Matrix

Example 7C#

Matrix M = new double[,]
{
    {5, -2, 3, 7 },
    {2, 1, -7, 3 },
    {4, 8, 9, 1 },
    {0, 5, -6, -3 }
};
Console.WriteLine($"Triu(M) = {Triu(M)}");
Console.WriteLine($"Tril(M) = {Tril(M)}");

OutputFrom the book

Triu(M) = 
 5  -2   3   7 
 0   1  -7   3 
 0   0   9   1 
 0   0   0  -3 

Tril(M) = 
 5   0   0   0 
 2   1   0   0 
 4   8   9   0 
 0   5  -6  -3