Hardware Visualized · YouTube · 1:41
This introductory linear algebra video demonstrates how to calculate the infinity norm and Euclidean norm for both vectors and matrices. Using a specific 3D vector and a 3x3 matrix as examples, it provides clear, step-by-step numerical derivations for each type of norm. The visual presentation highlights key components during calculations, making the definitions and procedures easy to follow for beginners.
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Generated from the video's visuals and explanation; not verbatim speech.
The video opens by presenting a specific mathematical problem: calculating the Euclidean and infinity norms for a given 3-dimensional vector and a 3x3 matrix A with elements [-3, 6, 8; 1, 5, 9; 4, 5, 6]. This sets the stage for demonstrating four distinct norm calculations.
The first concept introduced is the infinity norm of a matrix. The narrator explains that it is computed by taking the sum of the absolute values of the elements in each row, and then selecting the maximum of these row sums. Visually, as the calculation proceeds, each row of matrix A is highlighted in yellow when its absolute sum is being evaluated. The row sums are calculated as 17, 15, and 15, leading to the conclusion that the infinity norm of A is 17.
Next, the focus shifts to the infinity norm of a vector. The definition provided is the maximum absolute value among the vector's elements. For vector X, the absolute values of its components are calculated as 5, 7, and 9. A yellow box visually groups these absolute values before the maximum is taken, yielding an infinity norm of 9 for the vector.
The video then transitions to the Euclidean norm of a vector, which represents its magnitude. The procedure involves squaring each component of the vector, summing these squared values, and finally taking the square root of that sum. For vector X, this results in the calculation of the square root of 155, which is approximately 12.45.
Finally, the Euclidean norm of a matrix is explained. This is analogous to the vector case but applied to all elements of the matrix. Each element is squared, all squared values are summed, and the square root of the total sum is taken. For matrix A, this process yields the square root of 293, approximately 17.12. The video concludes after presenting all four calculated norms.
The infinity norm of a matrix is found by calculating the sum of the absolute values of the elements in each row, and then taking the maximum of these row sums. For the given matrix A, the row sums are 17, 15, and 15, so the infinity norm is 17.
The infinity norm of a vector is simply the maximum absolute value among all its components. For the vector , the absolute values are 5, 7, and 9, making the infinity norm 9.
Also known as the magnitude or L2 norm, the Euclidean norm of a vector is calculated by squaring each component, summing those squares, and taking the square root of the result. For vector X, this is √((-5)² + 7² + (-9)²) = √155 ≈ 12.45.
The Euclidean norm of a matrix (often called the Frobenius norm) is computed by squaring every element in the matrix, summing all those squared values, and taking the square root of the total. For matrix A, this results in √293 ≈ 17.12.
Explore conditions, steps and evidence. Supplementary explanations are labeled separately from content shown in the video.
The vector is displayed as in the problem statement and subsequent sections.
X
A given 3-dimensional vector with components -5, 7, and -9.
The matrix is displayed as in the problem statement and subsequent sections.
A
A given 3x3 matrix.
The infinity norm is denoted as ||A||_∞ for matrices and ||x||_∞ for vectors.
||·||_∞
The infinity norm (or maximum norm) of a matrix or vector.
Matrices or Vectors
The Euclidean norm is denoted as ||x||_e for vectors and ||A||_e for matrices.
||·||_e
The Euclidean norm (or L2 norm) of a matrix or vector.
Matrices or Vectors
The elements of matrix A are denoted as a_ij, where i is the row index and j is the column index.
a_{ij}
The element in the i-th row and j-th column of a matrix.
Real numbers
The components of vector x are denoted as , where i is the index.
The i-th component of a vector.
Real numbers
To compute the infinity norm of a matrix, calculate the sum of the absolute values of the elements in each row. The infinity norm is the largest of these row sums.
||A||_∞ = max(Σ|a_ij|) for
The infinity norm of a matrix is defined as the maximum of the sums of the absolute values of the elements in each row.
A is an n x n matrix
a_{ij} represents the element in the i-th row and j-th column of A
The infinity norm of a vector is defined as the maximum absolute value among its elements.
||x||_∞ = max(||) for
The infinity norm of a vector is the maximum absolute value among all its components.
x is an n-dimensional vector
represents the i-th component of the vector
To calculate the Euclidean norm of a vector, square each component of the vector, sum these squared values, take the square root of the result. This gives the magnitude of the vector.
||x||_e = sqrt() for to n
The Euclidean norm of a vector is the square root of the sum of the squares of its components, representing its magnitude.
x is an n-dimensional vector
represents the i-th component of the vector
To calculate the Euclidean norm of a matrix, square each element of the matrix, sum up all the squared values, take the square root of the sum. This results in the Euclidean norm of the matrix.
||A||_e = sqrt(Σ Σ a_ij^2) for to n, to n
The Euclidean norm of a matrix (also known as the Frobenius norm) is the square root of the sum of the squares of all its elements.
A is an n x n matrix
a_{ij} represents the element in the i-th row and j-th column of A
|-3| + |6| + |8| =
|1| + |5| + |9| =
|4| + |5| + |6| =
Row Sums : {17, 15, 15}
Infinity Norm of A is 17
Calculate the sum of absolute values for the first row.
Definition of row sum for infinity norm.
Calculate the sum of absolute values for the second row.
Definition of row sum for infinity norm.
Calculate the sum of absolute values for the third row.
Definition of row sum for infinity norm.
Find the maximum of the calculated row sums.
Definition of infinity norm as the maximum row sum.
The infinity norm of matrix A is 17.
|| = 5, || = 7, || = 9
||x||_∞ = max(5, 7, 9) = 9
Calculate the absolute value of each component of the vector.
Definition of absolute value.
Find the maximum of the absolute values.
Definition of infinity norm as the maximum absolute component.
The infinity norm of vector X is 9.
||x||_e = sqrt((-5)^)
||x||_e = sqrt()
||x||_e = sqrt(155)
||x||_e ≈ 12.45
Substitute the components of vector X into the Euclidean norm formula.
Definition of Euclidean norm.
Square each component.
Arithmetic calculation.
Sum the squared values.
Arithmetic calculation.
Take the square root of the sum.
Arithmetic calculation.
The Euclidean norm of vector X is approximately 12.45.
||A||_e = sqrt()
||A||_e = sqrt((-3)^)
||A||_e = sqrt()
||A||_e = sqrt(293)
||A||_e ≈ 17.12
Substitute the elements of matrix A into the Euclidean norm formula.
Definition of Euclidean norm for matrices.
Square each element of the matrix.
Arithmetic calculation.
Sum all the squared values.
Arithmetic calculation.
Take the square root of the sum.
Arithmetic calculation.
The Euclidean norm of matrix A is approximately 17.12.
Given: ,
Calculate the Euclidean norm and infinity norm.
Calculate the Euclidean norm and infinity norm for the given vector and matrix .
Vector
Matrix
Find ||X||_∞, ||X||_e, ||A||_∞, and ||A||_e.
Calculate the infinity norm of matrix A by finding the maximum row sum of absolute values.
Definition of matrix infinity norm.
Calculate the infinity norm of vector X by finding the maximum absolute component.
Definition of vector infinity norm.
Calculate the Euclidean norm of vector X by taking the square root of the sum of squared components.
Definition of vector Euclidean norm.
Calculate the Euclidean norm of matrix A by taking the square root of the sum of squared elements.
Definition of matrix Euclidean norm.
||A||_∞ = 17, ||X||_∞ = 9, ||X||_e ≈ 12.45, ||A||_e ≈ 17.12
The calculations follow the standard definitions of the respective norms and the arithmetic steps are shown explicitly in the video.
As each row's absolute sum is calculated, the corresponding row in the matrix is highlighted in yellow.
Matrix A
Row sums
Rows of matrix A are sequentially highlighted in yellow as their absolute sums are calculated.
The matrix elements themselves do not change.
The visual highlighting links the abstract formula for row sums directly to the specific rows being processed in the example matrix.
The absolute values of the vector components are enclosed in a yellow box before the maximum is taken.
Vector X
Absolute values
The calculated absolute values ||=5, ||=7, ||=9 are grouped and highlighted with a yellow box.
The original vector components remain unchanged.
This visual grouping emphasizes that the infinity norm operation is applied to the set of absolute values of the components.
The video presents the infinity norm for a matrix first, then for a vector, showing parallel definitions based on maximums.
While both use the concept of a 'maximum', the matrix infinity norm takes the maximum of row sums of absolute values, whereas the vector infinity norm takes the maximum of the absolute values of individual components.
The video presents the Euclidean norm for a vector, then extends the concept to a matrix by summing over all elements.
The Euclidean norm of a matrix (Frobenius norm) generalizes the Euclidean norm of a vector by treating the matrix as a flattened vector of all its elements and computing the square root of the sum of their squares.
To compute the infinity norm of a matrix, calculate the sum of the absolute values of the elements in each row. The infinity norm is the largest of these row sums.
The infinity norm of a vector is defined as the maximum absolute value among its elements.
To calculate the Euclidean norm of a vector, square each component of the vector, sum these squared values, take the square root of the result.
To calculate the Euclidean norm of a matrix, square each element of the matrix, sum up all the squared values, take the square root of the sum.
Covered · Black screen intro.
Covered · Problem statement displaying vector X and matrix A, asking for Euclidean and infinity norms.
Covered · Transition to Infinity Norm of a Matrix section.
Covered · Definition and step-by-step numerical derivation of the infinity norm for matrix A.
Covered · Transition to Infinity Norm of a Vector section.
Covered · Definition and step-by-step numerical derivation of the infinity norm for vector X.
Covered · Transition to Euclidean Norm of a Vector section.
Covered · Definition and step-by-step numerical derivation of the Euclidean norm for vector X.
Covered · Transition to Euclidean Norm of a Matrix section.
Covered · Definition and step-by-step numerical derivation of the Euclidean norm for matrix A.
Covered · Outro and thanks for watching.
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