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## Matrix Multiplication

Let be a general matrix and let denote a general matrix. Denote the matrix product by or . Then

matrix multiplicationis carried out by computing theinner productof every row of with every column of . Let theth row of be denoted by , , and theth column of by , . Then the matrix product is defined as

This definition can be extended tocomplexmatrices by using a definition of inner product which does not conjugate its second argument.^{7.4}

Examples:<!– MATH \begin{displaymath} \left[\begin{array}{cc} a & b \c & d \e & f \end{array}\right] \cdot## \left[\begin{array}{cc} \alpha & \beta \\gamma & \delta \end{array}\right]

\left[\begin{array}{cc} a\alpha+b\gamma & a\beta+b\delta

c\alpha+d\gamma & c\beta+d\delta

e\alpha+f\gamma & e\beta+f\delta \end{array}\right] \end{displaymath} –>

<!– MATH \begin{displaymath} \left[\begin{array}{cc} \alpha & \beta \\gamma & \delta \end{array}\right] \cdot## \left[\begin{array}{ccc} a & c & e \b & d & f \end{array}\right]

\left[\begin{array}{ccc} \alpha a + \beta b & \alpha c + \beta d & \alpha e + \beta f

\gamma a + \delta b & \gamma c + \delta d & \gamma e + \delta f \end{array}\right] \end{displaymath} –>

<!– MATH \begin{displaymath} \left[\begin{array}{c} \alpha \\beta \end{array}\right] \cdot## \left[\begin{array}{ccc} a & b & c \end{array}\right]

\left[\begin{array}{ccc} \alpha a & \alpha b & \alpha c

\beta a & \beta b & \beta c \end{array}\right] \end{displaymath} –>

An matrix can only be multiplied on the

rightby an matrix, where is any positive integer. An matrix can only be multiplied on theleftby a matrix, where is any positive integer. Thus, the number of columns in the matrix on the left must equal the number of rows in the matrix on the right.Matrix multiplication is

non-commutative, in general. That is, normally even when both products are defined (such as when the matrices are square.)The

transpose of a matrix productis the product of the transposes inreverse order:

The

identity matrixis denoted by and is defined as

Identity matrices are alwayssquare. The identity matrix , sometimes denoted as , satisfies for every matrix . Similarly, , for every matrix .As a special case, a matrix times a vector produces a new vector which consists of the inner product of every row of with

A matrix times a vector defines alinear transformationof . In fact, every linear function of a vector can be expressed as a matrix multiply. In particular, every linearfilteringoperation can be expressed as a matrix multiply applied to the input signal. As a special case, every linear, time-invariant (LTI) filtering operation can be expressed as a matrix multiply in which the matrix isToeplitz, i.e., (constant along alldiagonals).As a further special case, a row vector on the left may be multiplied by a column vector on the right to form a

single inner product:

where the alternate transpose notation “” is defined to includecomplex conjugationso that the above result holds also for complex vectors. Using this result, we may rewrite the general matrix multiply as