Matrix Computations: Basic Operations and Notation

Sources

• Golub, Gene H., and Charles F. Van Loan. Matrix computations. Vol. 3. JHU Press, 2012.
• Hogben, Leslie. Handbook of linear algebra. Chapman and Hall/CRC, 2013.

Vector Notation

• A (column) vector is an element of a finite vector space K^n with x = (x_i) and x_i \in K.

Basic Vector Operation

• Scalar Multiplication: K x K^n -> K^n <=> alpha * x = y
• Addition: K^n x K^n -> K^n <=> x + y = z
• Inner Product: K^n x K^n -> K <=> x^T * y = alpha
• Pointwise Multiplication and Division: K^n x K^n -> K^n <=> x.*y = z and x./y = z

Matrix Notation

• A matrix is an element of a finite vector space K^(m,n) with A = (a_ij) and a_ij \in K.

Basic Matrix Operation

• Transposition: K^(m,n) -> K^(n,m) <=> A^T = C
• Addition (Subtraction): K^(m,n) x K^(m,n) -> K^(m,n) <=> A+B = C
• Scalar Multiplication: K x K^(m,n) -> K^(m,n) <=> alpha * A = C
• Matrix-Matrix Multiplication: K^(m,k) x K^(k,n) -> K^(m,n) <=> A*B = C
• Pointwise Multiplication and Division : K^(m,n) x K^(m,n) -> K^(m,n) <=> A.*B = C and A./B = C

Additional Matrix and Vector Operations