




1. To multiply a matrix by a number you multiply every element of it by that number.
2. You may add two matrices which have the same shape: to do so add corresponding elements to get the corresponding element of the sum.
3. You may form the product of matrices A and B if the number of columns of A is the same as the number of rows of B. There is an entry in the product A B corresponding to each row of A and column of B: it is the dot product of the two: the sum of the products of corresponding terms.
Some definitions:
A matrix having only one row is called a row vector; a matrix having
only one column is called a column vector.
The transpose of a matrix A is the matrix, written as A^{T}
obtained by switching its rows and columns.
If A is n by m then A^{T} is m by n. (n by m always means
n rows and m columns.)
A matrix which is its own transpose is said to be symmetric.
If transposition changes the sign of every element of A then A
is said to be antisymmetric.
The square matrix having n ones on its diagonal and zeroes elsewhere, is called
the n dimensional identity matrix, written as I_{n} or
as I when its dimension is obvious. Multiplying I_{n}
on the right by any matrix A that has n rows yields: I_{n}
A = A.
The transpose of a column vector is a row vector.
A square matrix whose determinant is zero is said to be singular.
An n dimensional vector space is a space having at most n linearly independent
vectors. Any set of n linearly independent vectors in such a space forms a basis
for it. Any other vector in it is linearly dependent on the vectors in the basis.
The kth basis vector (row or column) is the row or column vector whose
components are all zeros except for a 1 in its kth place. Thus the 3rd
basis row vector in 4 dimensions is (0, 0, 1, 0).
A matrix defines the linear transformation which takes the kth
basis column vector into the kth column of the matrix. (This determines
the transformation completely by its linearity: its value on a linear combination
of basis vectors is the same linear combination of the columns of the matrix.)
The inverse of an n by n matrix A, is the matrix A^{1}
which obeys AA^{1}=I_{n}, and A^{1}A=I_{n}_{.}_{
}When n is finite, either one of these conditions implies
This is not always true in infinite dimensions: (consider the transformation T which takes the ith component of a vector and makes it the (i + 1)st with first component 0. Its inverse makes the (i+1)st into the ith. But what does this inverse do in acting on the vector v with a 1 as first component and all others 0? Since this vector is not in the range of T there is no way that applying T to T^{1}v to get this vector back.)