Section 16

Matrices

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A = (a_{ij})_{m \times n}

Rectangular array of scalars.

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(A + B)_{ij} = a_{ij} + b_{ij}

Entrywise sum for matrices of the same size.

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(c A)_{ij} = c a_{ij}

Scales every entry of the matrix.

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(A B)_{ij} = k \sum a_{ik} b_{k j}

Each entry arises from a row–column dot product.

The number of columns of $A$ must match the number of rows of $B$ . In general, $A B \neq = B A$ .

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A I = I A = A

Identity element for matrix multiplication.

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(A^{T})_{ij} = a_{j i}; (A B)^{T} = B^{T} A^{T}

Swaps rows and columns.

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A^{T} = A

Square matrix equal to its transpose.