(a-b)^2 Matrix

4 min read Jun 16, 2024

Understanding the (A-B)^2 Matrix

The expression (A-B)^2, where A and B are matrices, is a common operation in linear algebra. It represents the square of the difference between two matrices. However, unlike scalar algebra where squaring simply means multiplying a number by itself, matrix squaring requires a specific order of operations.

Expanding the Expression

The key to understanding (A-B)^2 is to expand it using the distributive property, just like in scalar algebra:

(A-B)^2 = (A-B)(A-B)

Now, we need to perform the matrix multiplication on the right-hand side.

Matrix Multiplication

Matrix multiplication follows specific rules:

Compatibility: The number of columns in the first matrix must equal the number of rows in the second matrix.
Resultant Matrix: The resulting matrix has the same number of rows as the first matrix and the same number of columns as the second matrix.
Element-wise Calculation: Each element of the resulting matrix is calculated by taking the dot product of a row from the first matrix and a column from the second matrix.

Applying Matrix Multiplication

Let's apply these rules to expand (A-B)^2:

(A-B)^2 = (A-B)(A-B)

= A(A-B) - B(A-B) (Using the distributive property)

= AA - AB - BA + BB

= A^2 - AB - BA + B^2

Note: Matrix multiplication is not commutative, meaning AB ≠ BA in general.

Example

Let's consider a simple example with 2x2 matrices:

A = [[1, 2], [3, 4]]

B = [[5, 6], [7, 8]]

Then:

(A-B)^2 = A^2 - AB - BA + B^2

A^2 = [[1, 2], [3, 4]] [[1, 2], [3, 4]] = [[7, 10], [15, 22]]

AB = [[1, 2], [3, 4]] [[5, 6], [7, 8]] = [[19, 22], [43, 50]]

BA = [[5, 6], [7, 8]] [[1, 2], [3, 4]] = [[23, 34], [31, 46]]

B^2 = [[5, 6], [7, 8]] [[5, 6], [7, 8]] = [[77, 90], [109, 128]]

Therefore,

(A-B)^2 = [[7, 10], [15, 22]] - [[19, 22], [43, 50]] - [[23, 34], [31, 46]] + [[77, 90], [109, 128]]

= [[32, 34], [3, 14]]

Conclusion

Understanding the (A-B)^2 matrix involves applying the distributive property and performing matrix multiplication carefully. The resulting matrix is not simply the square of the difference between individual elements of A and B. Remember that matrix multiplication is not commutative, so the order of operations matters.