Matrix-Matrix Multiplication

Matrix-Matrix Multiplication#

Composition of Linear Transformations#

If \(T_1: \mathbb{R}^n \to \mathbb{R}^m\) has matrix \(B\) and \(T_2: \mathbb{R}^m \to \mathbb{R}^\ell\) has matrix \(A\),
then the composition \(T_2 \circ T_1\) has matrix \(AB\), where:

\[ (AB)x = A(Bx). \]

The formula for matrix-matrix multiplication is:

\[ c_{ij} = \sum_{k} a_{ik} b_{kj} \]

where \(C = AB\).

Properties:

  • Defined only if the number of columns of \(A\) equals the number of rows of \(B\).

  • Associative: \((AB)C = A(BC)\).

  • Not commutative: \(AB \ne BA\) in general.

  • Transpose rule: \((AB)^T = B^T A^T\).

Identity and Inverse Matrices#

The identity matrix \(I_n\) has \(1\) on the diagonal and \(0\) elsewhere, and satisfies:

\[ I_n x = x \]

for all \(x \in \mathbb{R}^n\).

A square matrix \(A\) is invertible if there exists \(A^{-1}\) such that:

\[ A^{-1}A = AA^{-1} = I_n. \]

Fact

\(A\) is invertible if and only if its columns are linearly independent.

Change of Basis#

Let \(\{v_1, \dots, v_n\}\) be a basis of \(\mathbb{R}^n\). Any \(x \in \mathbb{R}^n\) can be written as:

\[ x = \sum_{i=1}^n \alpha_i v_i \]

where \(\alpha = (\alpha_1, \dots, \alpha_n)^T\) are the coordinates of \(x\) in this basis.

If \(V\) is the matrix with \(v_i\) as columns, then:

\[ x = V\alpha, \quad \alpha = V^{-1}x. \]

If \(A\) is the matrix of \(T\) in the standard basis and \(B\) is the matrix in the basis \(\{v_i\}\), then:

\[ A = V B V^{-1}. \]

This is the change of basis formula.

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