Matrix Flashcards
(16 cards)
Matrix scalar multiplication
Matrices can also be multiplied by a scalar. The rule is to just multiply every element of
the matrix. For example
Matrix multiplication
Matrices multiplied only if the number of columns of the
left matrix equals the number of rows of the right matrix. an m-by-n
matrix on the left can only be multiplied by an n-by-k matrix on the right. The resulting
matrix will be m-by-k. Evidently, matrix multiplication is generally not commutative
Commutative
An operation is said to be commutative if the order in which you apply it to two operands does not affect the result.
Addition is commutative because for any two numbers a and b, a + b = b + a. The order of adding the numbers doesn’t change the sum.
Example: 3 + 5 = 5 + 3 = 8
Example: 2 * 4 = 4 * 2 = 8
Subtraction and division, are not commutative:
Example: 7 - 3 ≠ 3 - 7 (7 - 3 = 4, but 3 - 7 = -4)
Example: 6 ÷ 2 ≠ 2 ÷ 6 (6 ÷ 2 = 3, but 2 ÷ 6 = 1/3)
Zero and Identity matrix
The zero matrix, denoted by 0, can be any size and is a matrix consisting of all zero
elements. Multiplication by a zero matrix results in a zero matrix.
The identity matrix, denoted by I, is a square matrix with ones down the main
diagonal
Multiplication by the identity matrix leaves the matrix unchanged
Diagonal matrix
A diagonal matrix has its only nonzero elements on the diagonal. diagonal matrices refer to square matrices, but they can also be rectangular
Band (or banded) matrix
Band matrix has nonzero elements only on diagonal bands. For example, a three-by-three band matrix with nonzero diagonals one above and one below a
nonzero main diagonal (called a tridiagonal matrix)
Upper or lower triangular matrix
An upper or lower triangular matrix is a square matrix that has zero elements below or
above the diagonal
Transpose of a matrix
The transpose of a matrix A, denoted by AT and spoken as A-transpose, switches the
rows and columns of A
Transpose of a matrix (details)
inner product (or dot product or scalar product) b
The inner product (or dot product or scalar product) between two vectors is obtained from
the matrix product of a row vector times a column vector. A row vector can be obtained
from a column vector by the transpose operator
Orthogonal vectors
If the inner product between two nonzero vectors is zero, we say that the vectors are
orthogonal. The norm of a vector is defined by
Outer product of matrix
For outter product of a matrxi every column is a multiple of the single vector u, and every row is a multiple
of the single vector vT
.
Inverse matrix
Square matrices may have inverses. When a matrix A has an inverse, we say it is invertible and denote its inverse by A−1
Conditions for Invertibility:
- The matrix must be square (i.e., it has the same number of rows and columns).
- The matrix must be non-singular, meaning its determinant is not zero det(A) != 0
Determinant of matrix
The determinant of a square matrix A is a scalar value that is calculated from the elements of the matrix.
For matrices larger than 2x2, the calculation involves summing the products of elements and their corresponding minors, often using a method known as Laplace expansion
Properties of Determinants
1.The determinant of a matrix is equal to the determinant of its transpose.
The determinant of a product of matrices is equal to the product of their determinants:
det(AB)=det(A)det(B).
- A matrix is invertible if and only if its determinant is non-zero.
- The determinant of an identity matrix is 1.
- If a matrix has a row or a column of zeros, its determinant is zero.
- Swapping two rows or columns of a matrix changes the sign of the determinant.
- Multiplying a row or column by a scalar multiplies the determinant by that scalar.
Inverse of matrix
