We have already seen several types of matrices, coefficient matrices, augmented matrices, and vectors. We have also discussed that we can view matrices as more general objects.
A matrix is a rectangular array of numbers called the entries, or elements, of the matrix.
The following are examples of matrices:
\left[ \begin{array}{cc} 1 & 2 \\ 2 & 4 \end{array} \right],\left[ \begin{array}{c} 1 \\ 2 \\ 4 \\ \sqrt{5} \end{array} \right], \left[ \begin{array}{ccc} 1 & 2 & 3 \\ 2 & 4 & -1 \end{array} \right], \left[ \begin{array}{cc} 0.2 & 2.3 \\ \pi & 0 \\ 0.001 & 3.2 \end{array} \right], \left[ \begin{array}{c} 4 \end{array} \right].
The size of the matrix is a description of the number of rows and columns it has. A matrix is called m \times n if it has m rows and n columns.
Thus if a coefficient matrix is an m \times n matrix, then the augmented matrix, which has one additional column, is an m \times (n+1) matrix. The above matrices have sizes 2 \times 2, 4 \times 1 , 2 \times 3 , 3 \times 2 and 1 \times 1.
A 1 \times n matrix is called a row matrix or a row vector. A m \times 1 matrix is called a column matrix or column vector.
We use a double script notation to refer to the entries of a matrix A. The entry of A in row i and column j is denoted by a_{ij}.
With this notation a general m \times n matrix A has the form:
A = \left[ \begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{array} \right]
If the columns of A are \vec{a}_1,...,\vec{a}_n, then we can write A as
A = \left[ \vec{a}_1 ... \vec{a}_n\right].
If the rows of A are \vec{A}_1,...,\vec{A}_m, then we can write A as
A = \left[ \begin{array}{c} \vec{A}_1\\ \vdots \\ \vec{A}_m \end{array} \right].
The diagonal entries of A are a_{11},a_{22},a_{33},....
If m = n then we say that A is a square matrix. A square matrix whose non-diagonal entries are all zero is called a diagonal matrix. A diagonal matrix whose diagonal entries are all the same is called a scalar matrix. If the scalar on the diagonal is 1, the scalar matrix is called the identity matrix.
Two matrices are equal if the have the same size and if their corresponding entries are equal. So A = [a_{ij}]_{m \times n} and B = [b_{ij}]_{r \times s} are equal if and only if m = r and n = s and a_{ij} = b_{ij} for all i and j.
Now that we have formally defined matrices, we can start examining their algebraic properties.
Addition and scalar multiplication work exactly as with vectors, just add and multiply component wise.
Notice that addition only makes sense if the matrices are of the same size.
Let A = [a_{ij}]. Then (-1)A=[-a_{ij}]. Thus A-A = [0_{ij}], the zero matrix. We will denote this matrix by O.
We can also define matrix multiplication:
If A is an m \times n matrix and B is an n \times r matrix, then the product C = AB is an m \times r matrix. The (i,j) entry of C is computed as follows c_{ij} = a_{i1}b_{1j}+a_{i2}b_{2j}+...+a_{in}b_{nj}.
Notice that c_{ij} is just the dot product of the ith row of A and the jth column of B, so in a sense, this product is a generalization of the dot product.
This corresponds nicely back to linear systems. Consider the system
\begin{array}{rrrrrrr} 3x & + & y & - & z & = & -2 \\ x & + & 3y & + & z & = & 0 \\ -x & + & y & + & 5z & = & 3 \end{array}
Observe that the left-hand side arises from the matrix product
\left[ \begin{array}{ccc} 3 & 1 & -1 \\ 1 & 3 & 1 \\ -1 & 1 & 5 \end{array}\right] \left[ \begin{array}{c} x \\ y \\ z \end{array} \right]
So the system can be written as
\left[ \begin{array}{ccc} 3 & 1 & -1 \\ 1 & 3 & 1 \\ -1 & 1 & 5 \end{array}\right] \left[ \begin{array}{c} x \\ y \\ z \end{array} \right] = \left[ \begin{array}{c} -2 \\ 0 \\ 3\end{array} \right]
Or A \vec{x} = \vec{b} where A is the coefficient matrix, \vec{x} is the (column) vector of variables, and \vec{b} is the (column) vector of constant terms.
Every linear system can be written in the form A \vec{x} = \vec{b}.
Thus A \vec{x} = \vec{b} has a solution if and only if \vec{b} is a linear combination of the columns of A.
We can also use multiplication of a matrix by vectors to "pick out" a column or row of a matrix. For this we use the special vectors e_i, which is the vector which is zero everywhere but the ith component, where it is 1.
Let A be an n \times m matrix, e_i a 1 \times m standard unit vector, and e_j an n \times 1 standard unit vector. Then
a. e_i A is the ith row of A and
b. Ae_j is the jth row of A.
It will often be convenient to regard a matrix as being composed of a number of smaller submatrices. By introducing vertical and horizontal lines into a matrix, we can partition it into blocks. For example
\left[ \begin{array}{ccccc} 1 & 0 & 0 & 3 & 4 \\ 0 & 1 & 0 & -2 & 0 \\ 0 & 0 & 1 & 4 & -1 \\ 0 & 1 & 2 & 0 & 0 \\ 3 & 4 & 1 & 0 & 0 \end{array} \right] = \left[\begin{array}{cc} I & B \\ C & O \end{array}\right]
where I is the 3 \times 3 identity matrix, B is 3 \times 2, C is 2 \times 3 and O is the 2 \times 2 zero matrix.
We can often speed up matrix multiplication by viewing a large matrix as many smaller matrices, via this partitioning into blocks.
Suppose A is an m \times n matrix and B is n \times r so that the product AB exists.
If we partition B in terms of its column vectors, as B = \left[ \vec{b}_1,...,\vec{b}_r \right] then
AB = A \left[ \vec{b}_1,...,\vec{b}_r \right] = \left[A\vec{b}_1,...,A\vec{b}_n\right].
This forms is called the matrix-column representation.
We can also do this with rows, but here we must do this with A instead of B. So if we partition A in terms of its row vectors,
then
This form is called the row-matrix representation of the product.
The definition of the matrix product AB uses the natural partition of A into rows and B into columns; this form is called the row-column representation of the product. We can also partition A into columns and B into rows; this form is called the column-row representation of the product. In this case we have
so
Now, if we want to work with block matrices, we can. We just view a block matrix as a big matrix with small matrices as entries.
For example, if we let A and B be matrices partitioned as
When A and B are two n \times n matrices, their product AB with also be an n \times n matrix.
If A = B it makes sense to define A^2 = AA and in general
A^k = A\cdots A where the right hand side is k factors.
Thus A^1 = A and we say that A^0 = I_n.
It is important to notice that we have the following properties:
1. A^r A^s = A^{r+s} and
2. (A^r)^s = A^{rs}.
So far, all of the matrix operations are analogous to operations on the real numbers, although they might not behave the same way. The next operation has no such analogue.
The transpose of an m \times n matrix A is the n \times m matrix A^T obtained by interchanging the rows and columns of A. That is, the ith column of A^T is the ith row of A for all i.
A square matrix A is symmetric if A^T = A--that is, if A is equal to its own transpose.
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