1.4: Matrices

Our usual notion of a matrix is that of a rectangular array of scalars. The definitions of matrix addition, multiplication, etc., are aimed at compactly representing and analyzing systems of equations of the form

\[\begin{aligned}
a_{11} x_{1}+\cdots+a_{1 n} x_{n} &=y_{1} \\
\cdots & \vdots \\
a_{m 1} x_{1}+\cdots+a_{m n} x_{n} &=y_{m}
\end{aligned}\]

This system of equations can be written as Ax = y if we define

\[A=\left(\begin{array}{ccc}
a_{11} & \cdots & a_{1 n} \\
\vdots & \cdots & \vdots \\
a_{m 1} & \cdots & a_{m n}
\end{array}\right), \quad x=\left(\begin{array}{c}
x_{1} \\
\vdots \\
x_{n}
\end{array}\right), \quad y=\left(\begin{array}{c}
y_{1} \\
\vdots \\
y_{m}
\end{array}\right)\]

The rules of matrix addition, matrix multiplication, and scalar multiplication of a matrix remain unchanged if the entries of the matrices we deal with are themselves (conformably dimensioned) matrices rather than scalars. A matrix with matrix entries is referred to as a block matrix or a partitioned matrix.

For example, the \(a_{i j}\), \(x_{j}\) and \(y_{i}\) in respectively A, x, and y above can be matrices, and P the equation Ax = y will still hold, as long as the dimensions of the various submatrices are conformable with the expressions \(\sum a_{i j} x_{j}=y_{i} \text { for } i=1, \cdots, m\) and \(j=1, \cdots, n\). What this requires is that the number of rows in \(a_{i j}\) should equal the number of rows in \(y_{i}\), the number of columns in \(a_{i j}\) should equal the number of rows in \(x_{j}\), and the number of columns in the \(x_{j}\) and \(y_{i}\) should be the same.

• Verify that

\[\left(\begin{array}{cc|c}
1 & 2 & 2 \\
0 & 1 & 3 \\
1 & 1 & 7
\end{array}\right)\left(\begin{array}{cc}
4 & 5 \\
8 & 9 \\
& \\
\hline & \\
2 & 0
\end{array}\right)=\left(\begin{array}{cc}
1 & 2 \\
0 & 1 \\
1 & 1
\end{array}\right)\left(\begin{array}{cc}
4 & 5 \\
8 & 9
\end{array}\right)+\left(\begin{array}{c}
2 \\
3 \\
7
\end{array}\right)\left(\begin{array}{cc}
2 & 0
\end{array}\right)\]

In addition to these simple rules for matrix addition, matrix multiplication, and scalar multiplication of partitioned matrices, there is a simple - and simply verified - rule for (complex conjugate) transposition of a partitioned matrix: if \([A]_{i j}=a_{i j}\), then \(\left[A^{\prime}\right]_{i j}=a_{j i}^{\prime}\), i.e., the (i, j)-th block element of \(A^{\prime}\) is the transpose of the (j, i)-th block element of A.

For more involved matrix operations, one has to proceed with caution. For instance, the determinant of the square block-matrix

\[A=\left(\begin{array}{ll}
A_{1} & A_{2} \\
A_{3} & A_{4}
\end{array}\right)\]

is clearly not \(A_{1} A_{4}-A_{3} A_{2}\) unless all the blocks are actually scalar! We shall lead you to the correct expression (in the case where \(A_{1}\) is square and invertible) in a future Homework.

Matrices as Linear Transformations

T is a transformation or mapping from X to Y , two vector spaces, if it associates to each \(x \in X\) a unique element \(y \in Y\). This transformation is linear if it satisfies

\[T(\alpha x+\beta y)=\alpha T(x)+\beta T(y)\]

• Verify that an n x m matrix A is a linear transformation from \(\mathbf{R}^{m}\) to \(\mathbf{R}^{n}\).

Does every linear transformation have a matrix representation? Assume that both X and Y are finite dimensional spaces with respective bases \(\left\{x_{1}, \ldots x_{m}\right\}\) and \(\left\{y_{1}, \ldots y_{n}\right\}\). Every \(x \in X\) can be uniquely expressed as: \(x=\sum_{i=1}^{m} a_{i} x_{i}\). Equivalently, every x is represented uniquely in terms of an element \(a \in \mathbf{R}^{m}\).Similarly every element \(y \in Y\) is uniquely represented in terms of an element \(b \in \mathbf{R}^{n}\). Now: T(\(x_{j}\)) = \(\sum_{i=1}^{n} b_{i j} y_{i}\) and hence

\[T(x)=\sum_{j=1}^{m} a_{j} T\left(x_{j}\right)=\sum_{i=1}^{n} y_{i}\left(\sum_{j=1}^{m} a_{j} b_{i j}\right)\]

A matrix representation is then given by B = (\(b_{i j}\)). It is evident that a matrix representation is not unique and depends on the basis choice.