Math Problem Statement

Consider following vectors � 1 2 � , � 2 3 � , � 3 3 � , � 4 5 � , � 5 6 � , � 6 5 �. What will be the normalized (i.e. unit length) principal component using PCA (choose option closest to your answer after rounding to third decimal digit)?

Solution

To find the normalized principal component of the given set of vectors using Principal Component Analysis (PCA), we need to go through the following steps:

  1. Construct the matrix: Arrange the given vectors as rows in a matrix.

    Given vectors: v1=[12],v2=[23],v3=[33],v4=[45],v5=[56],v6=[65]v_1 = \begin{bmatrix} 1 \\ 2 \end{bmatrix}, \quad v_2 = \begin{bmatrix} 2 \\ 3 \end{bmatrix}, \quad v_3 = \begin{bmatrix} 3 \\ 3 \end{bmatrix}, \quad v_4 = \begin{bmatrix} 4 \\ 5 \end{bmatrix}, \quad v_5 = \begin{bmatrix} 5 \\ 6 \end{bmatrix}, \quad v_6 = \begin{bmatrix} 6 \\ 5 \end{bmatrix}

    Combine these vectors to form matrix XX: X=[122333455665]X = \begin{bmatrix} 1 & 2 \\ 2 & 3 \\ 3 & 3 \\ 4 & 5 \\ 5 & 6 \\ 6 & 5 \end{bmatrix}

  2. Calculate the mean of each column and center the data by subtracting the mean from each value.

  3. Compute the covariance matrix of the centered data.

  4. Find the eigenvalues and eigenvectors of the covariance matrix. The eigenvector corresponding to the largest eigenvalue is the principal component.

  5. Normalize the principal component to have unit length.

I'll proceed to calculate this.The normalized (unit length) principal component, rounded to three decimal places, is approximately:

[0.7770.630]\begin{bmatrix} -0.777 \\ -0.630 \end{bmatrix}

Would you like further details on any of these steps or need assistance with related questions?

Here are some follow-up questions for practice:

  1. How does centering the data affect PCA results?
  2. Why is the principal component associated with the largest eigenvalue?
  3. What does the principal component represent in terms of variance?
  4. How would adding another vector impact the principal component?
  5. What is the geometric interpretation of a normalized principal component?

Tip: Normalizing vectors is essential in PCA, as it ensures that the principal component has a unit length, simplifying its interpretation as a direction rather than a magnitude.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Principal Component Analysis (PCA)
Vector Normalization
Eigenvalues and Eigenvectors

Formulas

Covariance matrix calculation
Eigenvalue decomposition
Normalization formula: ||v|| = sqrt(v1^2 + v2^2)

Theorems

Spectral Theorem for symmetric matrices

Suitable Grade Level

College/University Level