Lecture Notes in Differential Equations - Bruce E. Shapiro

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Definition 14.10. An m × n (or m by n) matrix A is a rectangular array
of number with m rows and n columns. We will denote the number in the
i th row and j th column as a ij
⎛
⎞
a 11 a 12 · · · a 1n
a 21 a 22 a 2n
A = ⎜
⎟
(14.12)
⎝ .
. ⎠
a m1 a m2 · · · a mn
We will sometimes denote the matrix A by [a ij ].
The transpose of the matrix A is the matrix obtained by interchanging
the row and column indices,
or
(A T ) ij = a ji (14.13)
[a ij ] T = [a ji ] (14.14)
The transpose of an m × n matrix is an n × m matrix. We will sometimes
represent the vector v = (x, y, z) by its 3×1 column-vector representation
⎛ ⎞
x
v = ⎝y⎠ (14.15)
z
or its 1 × 3 row-vector representation
v T = ( x y z ) (14.16)
Definition 14.11. Matrix Addition is defined between two matrices of
the same size, by adding corresponding elemnts.
⎛
⎞ ⎛
⎞ ⎛
⎞
a 11 a 12 · · · b 11 b 12 · · · a 11 + b 11 b 22 + b 12 · · ·
⎜
⎝
a 21 a 22 · · · ⎟ ⎜
⎠ + ⎝
b 21 b 22 · · · ⎟ ⎜
⎠ = ⎝
a 21 + b 21 a 22 + b 22 · · · ⎟
⎠
.
.
.
(14.17)
Matrices that have different sizes cannot be added.
Definition 14.12. A square matrix is any matrix with the same number
of rows as columns. The order of the square matrix is the number of rows
(or columns).
Definition 14.13. Let A be a square matrix. A submatrix of A is the
matrix A with one (or more) rows and/or one (or more) columns deleted.