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In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices.It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices that can be treated as single entities.
Matrices can be used to compactly write and work with multiple linear equations, that is, systems of linear equations. For example, if A is an m×n matrix, x designates a column vector (that is, n×1-matrix) of n variables x 1, x 2, ..., x n, and b is an m×1-column vector, then the matrix equation =
For a symmetric matrix A, the vector vec(A) contains more information than is strictly necessary, since the matrix is completely determined by the symmetry together with the lower triangular portion, that is, the n(n + 1)/2 entries on and below the main diagonal.
If the 4th component of the vector is 0 instead of 1, then only the vector's direction is reflected and its magnitude remains unchanged, as if it were mirrored through a parallel plane that passes through the origin. This is a useful property as it allows the transformation of both positional vectors and normal vectors with the same matrix.
The row space of this matrix is the vector space spanned by the row vectors. The column vectors of a matrix. The column space of this matrix is the vector space spanned by the column vectors. In linear algebra, the column space (also called the range or image) of a matrix A is the span (set of all possible linear combinations) of its column ...
A coordinate vector is commonly organized as a column matrix (also called a column vector), which is a matrix with only one column. So, a column vector represents both a coordinate vector, and a vector of the original vector space.
The cross product with respect to a right-handed coordinate system. In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here ), and is denoted by the symbol .
A vector specified as a row matrix is known as a row vector; one specified as a column matrix is known as a column vector. Again, an n-dimensional vector can be specified in either of the following forms using matrices: