α11 α12 ... α1n (matrix) α21 α22 ... α2n . . αij . . αm1 αm2 ... αmnwhere the α-s are real numbers (usually integers or rational numbers in these discussions), and the subscripts i,j,m,n are natural numbers, where i <= m, j <= n. The entry αij occupies the [i,j] place in the matrix, where i is the row number, and j is the column number. This is an mxn matrix, where m = the number of rows and n = the number of columns are its dimensions.
[1.1] (Equality) Two matrices are equal if and only if they have the same dimensions and all the corresponding entries are equal.
Notation: The equality
α11 α12 ... α1n β11 β12 ... β1k (matrix) α21 α22 ... α2n β21 β22 ... β2k . . αij . . = . . βij . . αm1 αm2 ... αmn βh1 βh2 ... βhkis true if and only if m=h, n=k and αij = βij for all natural nunmbers i <= m and j <= n.
One approach to the study of matrices involves a generalization of the idea of arrays as discussed in Volume D. A collection of 4 arrays array1 = (1,2,3), array2 = (9,8,7), array3 = (6,4,3), array4 = (3,7,2) may be collected into a 4x3 matrix as descending rows:
array1 1 2 3 (matrix) array2 = 9 8 7 array3 6 4 3 array4 3 7 2It may be more informative to use the term "row" in place of "array". If the matrix has a name A then it may be included in the row designations:
row1A 1 2 3 (matrix) A = row2A = 9 8 7 row3A 6 4 3 row4A 3 7 2The columns also form arrays: col1A = (1,9,6,3)t, col2A =(2,8,4,7)t, col3A = (3,7,3,2)t. The small superscript "t" resolves a contradiction in notation. Arrays (1,9,6,3), (2,8,4,7), (3,7,3,2) are written as horizontal rows. The "t" indicates that the arrays are to be transposed from horizontal to vertical notation:
col1A = col2A = col3A = (1,9,6,3)t = (2,8,4,7)t = (3,7,3,2)t = 1 2 3 9 8 7 6 4 3 3 7 2
The transpose of row arrays can be applied to matrices in general:
1 9 6 3 (matrix) At = 2 8 4 7 3 7 3 2
1 9 6 3 (matrix) At = 2 8 4 7 3 7 3 2
[2.1] (Transposition) The transpose of a matrix is a matrix obtained by interchanging all rows and all columns: the i-th row becomes the i-th column.
Notation: The entry in the [i,j] place of a matrix is moved to the [j,i] place in transpose of that matrix.
The j-th column becomes, after transposition, the j-th row. The transpose of an mxn matrix is an nxm matrix. The transpose interchanges the subscripts i,j that locate each entry of a matrix.
[2.2] (Transpose of a transpose) The transpose of the transpose of a matrix is the original matrix.
If the subscripts i,j that locate each entry of a matrix are interchanged twice, that is, [i,j] --> [j,i] -->[i,j], then i,j appear in the original order.
All the entries in a matrix form a rectangular pattern. In a square matrix they form a square. In that situation the number of rows equal the number of columns. The transposition of a square matrix (nxn) is again a square matrix (nxn). Some of the entries form diagonals of the square matrix.
[2.3] (Diagonals) The main diagonal contains all the entries on the line from the upper left to the lower right. The secondary diagonal contains all the entries on the line from the upper right to the lower left.
Notation: The main diagonal of an nxn matrix contains the entries in the locations [1,1], [2,2], ..., [n,n]. The secondary diagonal contains the entries at [1,n], [2,n-1], ..., [n,1].
For example, in the matrix
(matrix) 1 2 3 4 5 6 7 8 9the main diagonal contains the entries 1,5,9, and the secondary diagonal contains the entries 3,5,7.
It is quite possible that the matrix before transposition is identical to the matrix after transposition. For example,
(matrix) 8 1 4 2 1 6 7 5 4 7 3 9 2 5 9 0Notice equality of certain entries:
entry in place [i,j] above the main diagonal = entry in place [j,i] below the main diagonal 1 1 4 4 2 2 7 7 5 5 9 9The entries are symmetrical about the main diagonal 8,6,3,0.
[3.1] (Symmetric matrices) A square matrix is symmetric if the matrix after transposition is the same as the matrix before transposition.
Notation: A matrix is symmetric if and only if the entry in every [i,j] place is equal to the entry in the [j,i] place.
[3.2] (Diagonal matrices) A square matrix in which every entry off of the main diagonal is zero is a diagonal matrix.
Notation: A matrix is diagonal if and only if every entry in the [i,j] place is zero, where i and j are not equal.
Example:
1 0 0 0 (matrix) 0 2 0 0 0 0 3 0 0 0 0 4
0 and 1 play special roles in the number systems that were discussed in volume B.
[3.3] (Zero matrices) Any matrix in which all entries are zeros is called the zero matrix.
Notation: all [i,j] locations contain 0.
Example:
0 0 0 0 0 (matrix) O = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0is a 4x5 zero matrix. After matrix addition will have been defined, such matrices serve as additive identities.
[3.4] (Identity matrices) Any diagonal (square) matrix in which all entries along the main diagonal are 1 is an identity matrix.
Notation: All locations [i,j] contain 1 if i=j; all locations [i,j] contain 0 if i and j are not equal.
Example:
1 0 0 0 0 0 1 0 0 0 (matrix) I5 = 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1is a 5x5 identity matrix. After matrix multiplication will have been defined, such matrices serve as multiplicative identities.
[3.5a] (Upper triangular matrices) If all the entries below the main diagonal of a square matrix are zeros then the matrix is called an upper triangular matrix.
Notation: All locations [i,j] where i > j are zeros in an upper triangular matrix.
Example:
4 2 8 5 6 0 9 1 3 7 (matrix) 0 0 5 6 3 0 0 0 2 1 0 0 0 0 9
[3.5b] (Lower triangular matrices) If all the entries above the main diagonal of a square matrix are zeros then the matrix is called an lower triangular matrix.
Notation: All locations [i,j] where i < j are zeros in an upper triangular matrix.
Example:
4 0 0 0 0 2 9 0 0 0 (matrix) 8 1 5 0 0 5 3 6 2 0 6 7 3 1 9
It is obvious from the two examples that the transpose of an upper triangular matrix is a lower triangular matrix.