If both m and n for a matrix are equal to one, that matrix is called
a scalar (because it’s just a single value), and if either (but not both)
of m and n are one, that matrix may be called a vector. So in the
examples shown above, B and C are vectors; in some texts you’ll see
B called a column vector and C called a row vector. A square matrix
(that is, a matrix with m=n) with ones on the diagonal and zeroes in all
off-diagonal elements (such as matrix E above) is called the “unit” or
“identity” matrix, and a matrix with all elements equal to zero is called
the “null” or “zero” matrix.
Two matrices are said to be equal only if they have the same number
of rows as well as the same number of columns and if every element in
one matrix is equal in value to the corresponding element in the other
matrix. Matrices are usually denoted using uppercase letters or bold font,
often surrounded by square brackets (such as [A]), and the elements are
often written using lowercase letters with subscripts. So you may see the
elements of matrix [A] written as aij, although some authors use Aij or
[A]ij to refer to the elements of A (but be careful to note that Aij may
also refer to the “Matrix of Cofactors” of matrix A, which you can read
about in Section E of this document).
So does terminology such as “row vector” mean that matrices, vectors,
and tensors are the same thing? Not really. It’s certainly true that the
vectors and tensors in SGVT are often represented using matrices, but
it’s important to remember what those matrices stand for. Those arrays
of values represent the components of vectors and tensors, and those
components have meaning only when associated with the basis vectors
of a particular coordinate system. Since the basis vectors are not always
shown, it’s convenient to think of the matrix as representing the vector
or tensor itself, and that’s fine as long as you remember that the actual
vector or tensor has existence independent of any coordinate system.
A) Matrix addition, multiplication by a scalar, and subtraction
Matrices may be added only if both the row dimension (m) and their
column dimension (n) are equal (such matrices are said to be “of the
same order”). The addition is accomplished simply by adding each element
of one matrix to the corresponding element of the other matrix.
For example:
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