Linear Algebra

Determinants

Use the det function from numpy.linalg to compute the determinant of a matrix.

As shown in Figure 8-7, enter matrix data in cells A1:C3. Click cell F1, enter =PY(, and input:

a = xl("A1:C3").values  # Get matrix values(returns a DataFrame; use .values for the array)
np.linalg.det(a)         # Compute determinant

Press Ctrl+Enter. Cell F1 outputs the determinant of the first matrix (Figure 8-7).

For the second matrix (cells A5:C7, a singular matrix), click cell F5, enter =PY(, and input:

a = xl("A5:C7").values
np.linalg.det(a)

Press Ctrl+Enter. Cell F5 outputs 0 (since the matrix is singular) (Figure 8-7).

Figure 8-7 Calculating Matrix Determinants

Vector Operations

Vectors and matrices are mathematical concepts; arrays are computational concepts. A vector is a 1D array (row/column vector), and a matrix is a 2D array. Series and DataFrame are indexed 1D/2D arrays (structured arrays).

As shown in Figure 8-8, enter the first vector in A1:E1 and the second vector in A2:E2. Click cell B4, enter =PY(, and input:

x = xl("A1:E1")          # Get first vector(returns a DataFrame)
y = xl("A2:E2")          # Get second vector(returns a DataFrame)
x = x.values[0]           # Extract 1D array from DataFrame
y = y.values
x + y                     # Vector addition

Press Ctrl+Enter. Cell B4 outputs the sum of the vectors (Figure 8-8).

For vector subtraction, click cell B5, enter =PY(, and input:

x = xl("A1:E1").values
y = xl("A2:E2").values
x - y

Press Ctrl+Enter. Cell B5 outputs the difference (Figure 8-8).

For scalar multiplication (multiply the first vector by 10), click cell B6, enter =PY(, and input:

x = xl("A1:E1").values
x * 10

Press Ctrl+Enter. Cell B6 outputs the scaled vector (Figure 8-8).

For dot product, click cell B7, enter =PY(, and input:

x = xl("A1:E1").values
y = xl("A2:E2").values
np.dot(x, y)  # Dot product: sum of element-wise products

Press Ctrl+Enter. Cell B7 outputs the dot product (326) (Figure 8-8).

Figure 8-8 Vector Operations

Matrix Arithmetic

Matrix arithmetic includes addition, subtraction, multiplication, element-wise multiplication, division, and element-wise division.

As shown in Figure 8-9, enter the first matrix in A1:B2 and the second matrix in D1:E2.

Matrix Addition: Click cell B4, enter =PY(, and input:

x = xl("A1:B2").values
y = xl("D1:E2").values
np.add(x, y)  # Element-wise addition

Press Ctrl+Enter. Output: [[7 4], [13 9]] (Figure 8-9).

Matrix Subtraction: Click cell B5, enter =PY(, and input:

np.subtract(x, y)

Press Ctrl+Enter. Output: [[-5 2], [-5 3]] (Figure 8-9).

Element-wise Multiplication: Click cell B6, enter =PY(, and input:

np.multiply(x, y)  # Element-wise product

Press Ctrl+Enter. Output: [[6 3], [36 18]] (Figure 8-9).

Matrix Multiplication: Click cell B7, enter =PY(, and input:

np.dot(x, y)  # Matrix product

Press Ctrl+Enter. Output: [[33 10], [78 22]] (Figure 8-9).

Element-wise Division: Click cell B8, enter =PY(, and input:

np.divide(x, y)  # Element-wise division

Press Ctrl+Enter. Output: [[0.166..., 3.], [0.444..., 2.]] (Figure 8-9).

Matrix Division: Click cell B9, enter =PY(, and input:

z = np.linalg.inv(x)  # Inverse of x
np.dot(z, y)          # Equivalent to x⁻¹ · y(matrix division)

Press Ctrl+Enter. Output: [[-1.5 0.5], [2.5 0.166...]] (Figure 8-9).

Figure 8-9 Matrix Arithmetic

More Matrix Operations

This section covers matrix transposition, inversion, and reshaping.

As shown in Figure 8-10, enter the first matrix in A1:C3 and the second matrix in E1:H3.

Transposition: Click cell B5, enter =PY(, and input:

a = xl("A1:C3").values
np.transpose(a)  # Transpose the matrix

Press Ctrl+Enter. Output: [[1 2 4], [2 5 10], [0 -1 -1]] (Figure 8-10).

Inversion: Click cell B6, enter =PY(, and input:

np.linalg.inv(a)  # Inverse of the matrix

Press Ctrl+Enter. Output: [[5 2 -2], [-2 -1 1], [0 -2 1]] (Figure 8-10).

Reshaping: Click cell B7, enter =PY(, and input:

a = xl("E1:H3").values
np.reshape(a, [2, 6])  # Reshape to 2 rows × 6 columns

Press Ctrl+Enter. Output: [[1 4 7 10 2 5], [8 11 3 6 9 12]] (Figure 8-10).

Figure 8-10 Matrix Transposition, Inversion, and Reshaping

Matrix Decomposition

This section covers LU decomposition and QR decomposition:

LU decomposition: Factorizes a matrix into upper/lower triangular matrices (basis for Gaussian elimination).

QR decomposition: Factorizes a matrix into an orthogonal matrix

and an upper triangular matrix

.

As shown in Figure 8-11, enter the matrix in A1:C3.

LU Decomposition: Click cell B5, enter =PY(, and input:

import scipy as sp
a = xl("A1:C3").values
r = sp.linalg.lu(a)  # LU decomposition

Press Ctrl+Enter. Output: Tuple containing permutation matrix

, lower triangular matrix

, and upper triangular matrix

(Figure 8-11).

QR Decomposition: Click cell B6, enter =PY(, and input:

r = sp.linalg.qr(a)  # QR decomposition

Press Ctrl+Enter. Output: Tuple containing orthogonal matrix

and upper triangular matrix

(Figure 8-11).

Figure 8-11 Matrix Decomposition