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% A column vector 4 by 1, with three numbers you fill in by yourself
colVec = [5;2;3;10]
colVec = 4x1
5
2
3
10
% Another column vector with 4 random numbers
colVecRand = rand(4,1)
colVecRand = 4x1
0.4899
0.1679
0.9787
0.7127
% A row vector 1 by 4
rowVec = [3,2,4,5]
rowVec = 1x4
3 2 4 5
% A row vector 1 by 4 with random number
rowVecRand = rand(1,4)
rowVecRand = 1x4
0.5005 0.4711 0.0596 0.6820% A 2 by 3 matrix by hand
matA = [1,2,1;
3,4,10]
matA = 2x3
1 2 1
3 4 10
% Another 2 by 3 matrix, now with random numbers
matRand = rand(2,3)
matRand = 2x3
0.0424 0.5216 0.8181
0.0714 0.0967 0.8175
% Another 2 by 3 matrix, now with random integers between 1 and 10
% rand draws between 0 and 1, ceil converts 0.1 to 1, 1.1 to 2, etc
matRand = ceil(rand(2,3)*10)
matRand = 2x3
8 7 10
2 6 7% A 4 by 4 square matrix
matSquare = rand(4)
matSquare = 4x4
0.8003 0.0835 0.8314 0.5269
0.4538 0.1332 0.8034 0.4168
0.4324 0.1734 0.0605 0.6569
0.8253 0.3909 0.3993 0.6280
% or can define 4 by 4
matSquare = rand(4, 4)
matSquare = 4x4
0.2920 0.1672 0.4897 0.0527
0.4317 0.1062 0.3395 0.7379
0.0155 0.3724 0.9516 0.2691
0.9841 0.1981 0.9203 0.4228
% or can define 4 by 4, between 1 and 5 each number
matSquare = ceil(rand(4, 4)*5)
matSquare = 4x4
3 2 4 5
5 4 4 1
3 4 1 1
5 3 1 3If a matrix \(A\) is square matrix with the same number of rows and columns, and all diagonal elements are \(1\) and non-diagonal elements are \(0\), then \(A\) is an identity matrix:
\(A_{i,j}\) are the value in the ith row and jth column of the matrix \(A\)
\(A\) is an identity matrix, when: \(A_{i,j} =0\;\textrm{if}\;i\not= j\), \(A_{i,j} =1\;\textrm{if}\;i=j\)
% 4 by 4 identity matrix
identity4by4 = eye(4)
identity4by4 = 4x4
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1When a matrix is muplieid by the identity matrix, you get the same matrix back, for example, multiplying random integer 4 by 4 matrix by the 4 by 4 identity matrix:
matSquare
matSquare = 4x4
3 2 4 5
5 4 4 1
3 4 1 1
5 3 1 3
matSquareTimesIdentity = matSquare*identity4by4
matSquareTimesIdentity = 4x4
3 2 4 5
5 4 4 1
3 4 1 1
5 3 1 3When a row vector is muplieid by the identity matrix, you get the same vector back, for example, multiplying random integer 1 by 4 row vector by the 4 by 4 identity matrix:
rowVec
rowVec = 1x4
3 2 4 5
rowVecTimesIdentity = rowVec*identity4by4
rowVecTimesIdentity = 1x4
3 2 4 5When an identity matrix is multiplied by a column vector, you get the same vector back, for example, multiplying 4 by 4 identity matrix by random integer 4 by 1 column vector by the :
colVec
colVec = 4x1
5
2
3
10
colVecTimesIdentity = identity4by4*colVec
colVecTimesIdentity = 4x1
5
2
3
10A lower triangular matrix is a square matrix where:
Square matrix\(A\) is a lower triangular matrix, when: \(A_{i,j} =0\;\textrm{if}\;i<j\)
Square matrix\(A\) is a upper triangular matrix, when: \(A_{i,j} =0\;\textrm{if}\;i>j\)
% lower triangular matrix of matA
lowerTriangular = tril(matSquare)
lowerTriangular = 4x4
3 0 0 0
5 4 0 0
3 4 1 0
5 3 1 3
% upper triangular matrix of matA
upperTriangular = triu(matSquare)
upperTriangular = 4x4
3 2 4 5
0 4 4 1
0 0 1 1
0 0 0 3% 3 by 3 by 2, storing multiple matrixes together in tenA
tenA = zeros(3,3,2);
tenA(:,:,1) = rand(3,3);
tenA(:,:,2) = rand(3,3);
disp(tenA);
(:,:,1) =
0.8819 0.3689 0.1564
0.6692 0.4607 0.8555
0.1904 0.9816 0.6448
(:,:,2) =
0.3763 0.4820 0.2262
0.1909 0.1206 0.3846
0.4283 0.5895 0.5830
% Creating four 2 by 3 matrixes
matRand = rand(2,3,4)
matRand =
matRand(:,:,1) =
0.2518 0.6171 0.8244
0.2904 0.2653 0.9827
matRand(:,:,2) =
0.7302 0.5841 0.9063
0.3439 0.1078 0.8797
matRand(:,:,3) =
0.8178 0.5944 0.4253
0.2607 0.0225 0.3127
matRand(:,:,4) =
0.1615 0.4229 0.5985
0.1788 0.0942 0.4709
disp(matRand);
(:,:,1) =
0.2518 0.6171 0.8244
0.2904 0.2653 0.9827
(:,:,2) =
0.7302 0.5841 0.9063
0.3439 0.1078 0.8797
(:,:,3) =
0.8178 0.5944 0.4253
0.2607 0.0225 0.3127
(:,:,4) =
0.1615 0.4229 0.5985
0.1788 0.0942 0.4709