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# Matrices

A matrix is a rectangular collection of numbers (again, possibly complex). It might represent a collection of row/column vectors, or a transformation that changes a vector into a different vector. For example, the rotation of a real 2D vector by an angle about the axis could be represented as a 2x2 matrix,

The effect of this rotation on a 2D vector in the plane can be determined by multiplying the vector by this transformation matrix,
 (9)

where and .

More generally, the product of a matrix and a column vector may be written as

 (10)

where
 (11)

Notice that each element of the product, , is just the dot product of the th row of the matrix with the vector. It is also possible to think of the column vector as being a linear combination of the columns of the matrix , with each column having a weight .

It is also possible to left-multiply'' a matrix by a row vector, like this:

 (12)

where
 (13)

This time, each element of the product, , is the dot product of the vector on the left with the th column of the matrix. The row vector may be thought of as a linear combination of the rows of the matrix , with each row having a weight of .

From the information given, it should be obvious that a row vector times a matrix times a column vector yields a number (scalar).

Now consider the multiplication of two matrices and , which yields a new matrix, :

 (14)

where
 (15)

That is, each element of the product matrix is the result of a dot product between row of the matrix and column of the matrix . It is possible to think of matrix multiplication as a generalization of the product of a matrix times a vector, where now instead of one column vector on the right, we have a series of them. It is also possible to think of matrix multiplication as a generalization of the product of a row vector times a matrix, where now instead of one row vector on the left, we have a series of them.

The transpose of a matrix , typically denoted , is obtained simply by swapping elements across the diagonal, . For example,

 (16)

For complex numbers, usually the complex conjugate transpose, or adjoint, is more useful. The adjoint of a matrix is often denoted . For example,

 (17)

If a matrix is equal to its adjoint , it is said to be a Hermitian matrix. Clearly, this can only happen if the diagonal elements are real (otherwise, will never equal , for example). One can also take the adjoint of a vector, which is effectively what one does in order to take a dot product of two complex vectors (see above):
 (18)

Next: Determinants Up: linear_algebra Previous: Vectors
David Sherrill 2006-08-15