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Matrices

Matrices. Definitions. A matrix is an m x n array of scalars, arranged conceptually as m rows and n columns. m is referred to as the row dimension n is referred to as the column dimension If m = n , the matrix is a square matrix. . Representations.

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Matrices

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  1. Matrices

  2. Definitions • A matrix is an m x n array of scalars, arranged conceptually as m rows and n columns. • m is referred to as the row dimension • n is referred to as the column dimension • If m=n, the matrix is a square matrix.

  3. Representations • Each element of array A is represented as: • Array A can thus be represented as: • The transpose of A is: • The column matrix of A is: • The corresponding row matrix is:

  4. Matrix Operations • Scalar-matrix multiplication • Matrix-matrix addition: The sum makes sense only if the two matrices have the same dimensions. • Matrix-matrix multiplication The matrix-matrix product is defined only if the number of columns of A is the same as the number of rows of B.

  5. Operation Properties

  6. Row-Column Matrices vs. Transpose • We may represent any point in a space as a row or column matrix (or vector). • Transpose

  7. Inverse • Matrix A is invertible if there exists a B such that: AB = I • Such matrix A is said to be nonsingular and B can denoted by A-1. • The inverse of a square matrix A exists if and only if |A|, determinant of A, is nonzero.

  8. Identity Matrix • The identity matrix I is a square matrix with 1’s on the diagonal and 0’s elsewhere: • AI = A, IB = B

  9. Rank The row (column) rank is the maximum number of linearly independent rows (columns).

  10. Rank (II) • For an n x n matrix, if it is nonsingular, i.e., both of its row rank and column rank are n, the matrix has rank of n.

  11. Basis Transformations

  12. Cross Product

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