Introduction Flashcards
Column vector
A d × 1 matrix
Row vector
A 1 × d matrix
The dot product is a commutative operation
Squared norm or Euclidean norm
Lp-norm
The (squared) Euclidean distance between x and y
Dot products satisfy the Cauchy-Schwarz inequality, according to which the dot product between a pair of vectors is bounded above by the product of their lengths
The cosine function between two vectors
The cosine law
You are given the orthonormal directions [3/5, 4/5] and [−4/5, 3/5]. One can represent the point [10, 15] in a new coordinate system defined by the directions [3/5, 4/5] and [−4/5, 3/5] by computing the dot product of [10, 15] with each of these vectors.
Therefore, the new coordinates [x’, y’] are defined as follows:
x’ = 10 ∗ (3/5) + 15 ∗ (4/5) = 18
y’ = 10 ∗ (−4/5) + 15 ∗ (3/5) = 1
One can express the original vector using the new axes and coordinates as follows:
[10, 15] = x’[3/5, 4/5] + y’[−4/5, 3/5]
An example of a multiplication of a 3×2 matrix A = [aij] with a 2-dimensional column vector
x = [x1, x2]T
An example of the multiplication of a 3-dimensional row vector v = [v1, v2, v3] with the 3 × 2 matrix A
The multiplication of an n×d matrix A with a d-dimensional column vector x to create an n-dimensional column vector Ax is a weighted sum
Outer Product
The outer product is not commutative; the order of the operands matters
An example of a matrix multiplication
Types of matrices
Inverse of a 2 x 2 matrice
An orthogonal matrix is a square matrix whose inverse is its transpose
Infinite geometric series of matrices
Frobenius Norm (energy)
Defined as the square root of the sum of the absolute squares of the elements of a matrix
The trace of a square matrix A
Is defined by the sum of its diagonal entries
The energy of a rectangular matrix A is equal to the trace of either AAT or AT A
Pre-multiplying a matrix X with an elementary matrix corresponding to an interchange results in an interchange of the rows of X