Matrix Operation
A Matrix Operation is a mathematical operation whose operands include a Matrix.
- Context:
- It can range from being an Abstract Matrix Operation to being a Matrix Data Structure Operation.
- It can range from being a Unary Matrix Operation, to being a Binary Matrix Operation to being an Complex Matrix Operation.
- It can be supported by a Matrix Processing System.
- It can be a Vector Operation, ranging from being a Matrix Column-wise Operation to being a Matrix Row-wise Operation.
- Example(s):
- Matrix Multiplication Operation (matrix-matrix multiply, or matrix-vector multiply).
- Matrix Transpose Operation.
- Matrix Conjugate Transpose Operation.
- Matrix Factorization.
- Matrix Inverse Operation, for a Matrix Inverse.
- Matrix Determinant Operation, for a Matrix Determinant.
- Matrix Trace Operation, for a Matrix Trace.
- Matrix Rank Operation, for a Matrix Rank.
- …
- Counter-Example(s):
- See: Linear Algebra Task, Linear Algebra Library.
References
2015
- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/comparison_of_linear_algebra_libraries Retrieved:2015-2-1.
- The following tables provide a comparison of linear algebra software libraries, either specialized or general purpose libraries with significant linear algebra coverage. ...
- Operations:
- TF - triangular factorizations (LU, Cholesky)
- OF - orthogonal factorizations (QR, QL, generalized factorizations)
- EVD - eigenproblems.
- SVD - singular value decomposition.
- GEVD - generalized EVD.
- GSVD - generalized SVD
2013
- http://www.mathworks.com/help/matlab/matlab_prog/array-vs-matrix-operations.html
- QUOTE: Array operations work on corresponding elements of arrays with equal dimensions. For vectors, matrices, and multidimensional arrays, both operands must be the same size. Each element in the first input gets matched up with a similarly located element from the second input. If the inputs are different sizes, MATLAB cannot match the elements one-to-one.
As a simple example, you can add two vectors with the same length. …
…
… Matrix operations follow the rules of linear algebra and are not compatible with multidimensional arrays. The required size and shape of the inputs in relation to one another depends on the operation. For nonscalar inputs, the matrix operators generally calculate different answers than their array operator counterparts.
- QUOTE: Array operations work on corresponding elements of arrays with equal dimensions. For vectors, matrices, and multidimensional arrays, both operands must be the same size. Each element in the first input gets matched up with a similarly located element from the second input. If the inputs are different sizes, MATLAB cannot match the elements one-to-one.
2012
- Are matrices and second rank tensors the same thing?
- QUOTE: Matrices are often first introduced to students to represent linear transformations taking vectors from [math]\displaystyle{ ℝ^n }[/math] and mapping them to vectors in [math]\displaystyle{ ℝ^m }[/math]. A given linear transformation may be represented by infinitely many different matrices depending on the basis vectors chosen for [math]\displaystyle{ ℝ^n }[/math] and [math]\displaystyle{ ℝ^m }[/math], and a well-defined transformation law allows one to rewrite the linear operation for each choice of basis vectors.
Second rank tensors are quite similar, but there is one important difference that comes up for applications in which non-Euclidean (non-flat) distance metrics are considered, such as general relativity. 2nd rank tensors may map not just ℝn to ℝm, but may also map between the dual spaces of either [math]\displaystyle{ ℝ^n }[/math] or [math]\displaystyle{ ℝ^m }[/math]. The transformation law for tensors is similar to the one first learned for linear operators, but allows for the added flexibility of allowing the tensor to switch between acting on dual spaces or not.
Note that for Euclidean distance metrics, the dual space and the original vector space are the same, so this distinction doesn't matter in that case.
- QUOTE: Matrices are often first introduced to students to represent linear transformations taking vectors from [math]\displaystyle{ ℝ^n }[/math] and mapping them to vectors in [math]\displaystyle{ ℝ^m }[/math]. A given linear transformation may be represented by infinitely many different matrices depending on the basis vectors chosen for [math]\displaystyle{ ℝ^n }[/math] and [math]\displaystyle{ ℝ^m }[/math], and a well-defined transformation law allows one to rewrite the linear operation for each choice of basis vectors.