Construction of Matrices: Mathematica Programs

Construction of Matrices

1. Identity Matrix: Construct an identity matrix of size 4.

IdentityMatrixConstruction[n_] := IdentityMatrix[n]
IdentityMatrixConstruction[4]

Output:

{{1, 0, 0, 0},

 {0, 1, 0, 0},

 {0, 0, 1, 0},

 {0, 0, 0, 1}}

2. Diagonal Matrix

Construct a diagonal matrix with diagonal elements {1, 2, 3, 4}.

DiagonalMatrixConstruction[diagonalElements_List] := DiagonalMatrix[diagonalElements]
DiagonalMatrixConstruction[{1, 2, 3, 4}]

Output:

{{1, 0, 0, 0},

 {0, 2, 0, 0},

 {0, 0, 3, 0},

 {0, 0, 0, 4}}

3. Random Matrix

Generate a 3×4 matrix of random integers between 0 and 10.

RandomMatrixConstruction[n_, m_] := RandomInteger[{0, 10}, {n, m}]

RandomMatrixConstruction[3, 4]

Example Output:

{{3, 8, 1, 7},

 {9, 0, 5, 4},

 {6, 2, 10, 8}}

4. Zero Matrix

Create a zero matrix of size 3×3.

ZeroMatrixConstruction[n_, m_] := ConstantArray[0, {n, m}]

ZeroMatrixConstruction[3, 3]

Output:

{{0, 0, 0},

 {0, 0, 0},

 {0, 0, 0}}

5. Upper Triangular Matrix

Generate an upper triangular matrix from the given matrix.

UpperTriangularMatrixConstruction[mat_List] := UpperTriangularize[mat]

UpperTriangularMatrixConstruction[{{1, 2, 3}, {4, 5, 6}, {7, 8, 9}}]

Output:

{{1, 2, 3},

 {0, 5, 6},

 {0, 0, 9}}

6. Lower Triangular Matrix

Generate a lower triangular matrix from the given matrix.

LowerTriangularMatrixConstruction[mat_List] := LowerTriangularize[mat]

LowerTriangularMatrixConstruction[{{1, 2, 3}, {4, 5, 6}, {7, 8, 9}}]

Output:

{{1, 0, 0},

 {4, 5, 0},

 {7, 8, 9}}

7. Symmetric Matrix

Generate a symmetric matrix from the given matrix by averaging it with its transpose.

SymmetricMatrixConstruction[mat_List] := (mat + Transpose[mat])/2

SymmetricMatrixConstruction[{{1, 2}, {3, 4}}]

Output:

{{1, 2.5},

 {2.5, 4}}

8. Block Matrix

Create a block matrix from submatrices.

BlockMatrixConstruction[A_, B_, C_, D_] := ArrayFlatten[{{A, B}, {C, D}}]

BlockMatrixConstruction[{{1, 2}, {3, 4}}, {{5, 6}}, {{7, 8}}, {{9}}]

Output:

{{1, 2, 5, 6},

 {3, 4, 5, 6},

 {7, 8, 9, 9}}

9. Toeplitz Matrix

Generate a Toeplitz matrix from the first row and first column.

ToeplitzMatrixConstruction[firstRow_List, firstCol_List] := ToeplitzMatrix[firstRow, firstCol]

ToeplitzMatrixConstruction[{1, 2, 3}, {1, 4, 5}]

Output:

{{1, 2, 3},

 {4, 1, 2},

 {5, 4, 1}}

10. Hilbert Matrix

Generate a 3×3 Hilbert matrix.

HilbertMatrixConstruction[n_] := Table[1/(i + j – 1), {i, n}, {j, n}]

HilbertMatrixConstruction[3]

Output:

{{1, 1/2, 1/3},

 {1/2, 1/3, 1/4},

 {1/3, 1/4, 1/5}}