Matrices in Scilab

Scilab is a powerhouse in the world of matrix manipulation. In this comprehensive guide, we'll explore how to craft a matrix from scratch and execute pivotal matrix operations with precision.

First and foremost, let's demystify matrices. A matrix, at its core, is a two-dimensional array of numbers. Consider this matrix with two rows and three columns for clarity: $$ M = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix} $$ Such matrices are the cornerstone of numerous advanced scientific and engineering applications.

Creating a matrix in Scilab is straightforward:

• Wrap your matrix elements in square brackets.
• Separate elements within the same row using commas or spaces.
• Designate a new row with a semicolon.

For illustrative purposes, here's a 3x3 matrix:

M = [1, 2, 3;
4, 5, 6;
7, 8, 9]

Alternatively, you can also write using spaces instead of commas.

M = [1 2 3;
4 5 6;
7 8 9]

This matrix comprises nine elements, systematically arranged in a three-by-three configuration.

The comma-separated numbers denote members of the same row, while the semicolon (`;`) flags the inception of a subsequent row.

$$ M = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} $$

Once your matrix is primed, you're equipped to undertake an array of matrix calculations.

Addition and Subtraction

Matrices can be seamlessly added or subtracted provided they align in dimensions.

Let's initiate with two 2x2 matrices:

A = [1, 2;
3, 4]

B = [4, 5;
6, 7]

Combine them with A+B

A + B

This results in a 2x2 matrix, where each element is the sum of its counterparts:

ans=
6. 8.
10. 12.

For a clearer perspective:

$$ A+B = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} + \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} = \begin{pmatrix} 1+5 & 2+6 \\ 3+7 & 4+8 \end{pmatrix} = \begin{pmatrix} 6 & 8 \\ 10 & 12 \end{pmatrix} $$

Matrix Multiplication

To multiply one matrix by another, ensure that the column count of the first matrix coincides with the row count of the second.

Take, for example, these matrices:

A = [1, 2;
3, 4]

B = [5, 6;
7, 8]

Execute a row-by-column multiplication

A*B

The result unfurls as a 2x2 matrix

ans =
19. 22.
43. 50.

To elucidate:

$$ A*B = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} * \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} = \begin{pmatrix} 1 \cdot 5 + 2 \cdot 7 & 1 \cdot 6 + 2 \cdot 8 \\ 3 \cdot 5 + 4 \cdot 7 & 3 \cdot 6 + 4 \cdot 8 \end{pmatrix} = \begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix} $$

Scalar Multiplication

In scalar multiplication, every matrix element gets amplified by a predetermined scalar.

For a 2x2 matrix:

A = [1, 2;
3, 4]

Double it

2*A

The product? A 2x2 matrix with elements duplied.

ans =
2. 4.
6. 8.

To elucidate further:

$$ 2*A = 2 \cdot \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} = \begin{pmatrix} 2 \cdot 1 & 2 \cdot 2 \\ 2 \cdot 3 & 2 \cdot 4 \end{pmatrix} = \begin{pmatrix} 2 & 4 \\ 6 & 8 \end{pmatrix} $$

Matrix Functions in Scilab

Beyond the rudiments, Scilab proffers an array of dedicated functions tailored for matrix computations. Noteworthy mentions include:

• det(A)
  Computes the determinant of matrix A.
• inv(A)
  Retrieves the inverse of matrix A, when feasible.
• eye(n, m)
  Constructs an nxm identity matrix.
• size(A)
  Fetches the dimensions of matrix A.

Stay tuned. Subsequent lessons will unravel Scilab's full suite of matrix-oriented capabilities.