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Implement matrix transforms and discuss eigenvalues

GoogleSoftware Engineer · Onsite
MediumSoftware EngineerOnsiteSoftware Engineering Fundamentals

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This problem tests your ability to perform in-place 2D matrix transformations and your conceptual understanding of how linear algebraic properties like eigenvalues describe matrix behavior.

Part 1: Coding

Given an n x n 2D matrix representing an image, rotate the image by 90 degrees clockwise in-place. You must modify the input matrix directly without allocating a second matrix.

Function Signature

def rotate(matrix: List[List[int]]) -> None:

Constraints

  • n == matrix.length == matrix[i].length
  • 1 <= n <= 500
  • -1000 <= matrix[i][j] <= 1000

Examples

Example 1: Input: matrix = [[1, 2], [3, 4]] Output: [[3, 1], [4, 2]]

Example 2: Input: matrix = [[5, 1, 9, 11], [2, 4, 8, 10], [13, 3, 6, 7], [15, 14, 12, 16]] Output: [[15, 13, 2, 5], [14, 3, 4, 1], [12, 6, 8, 9], [16, 7, 10, 11]]

Part 2: Conceptual Discussion

  1. Explain what an eigenvalue and eigenvector represent in the context of a linear transformation matrix A.
  2. If a matrix is used to represent a coordinate transformation, what does a large eigenvalue (relative to others) tell you about how the transformation behaves along the direction of its corresponding eigenvector?
  3. How would you store a matrix that is 99% filled with zeros efficiently, and how does this storage affect matrix-vector multiplication performance?

Hints

Hint 1

To perform an in-place 90-degree rotation, consider splitting the operation into two steps: a transposition followed by a row reversal.

Hint 2

The eigenvalue equation `Av = λv` implies that vectors scaled by `λ` do not change direction under transformation `A`.

Hint 3

For sparse matrices, avoid standard 2D arrays and consider using coordinate formats (COO) or Compressed Sparse Row (CSR) formats to store only non-zero entries.

Follow-up questions

  • How would you modify your rotation algorithm if the matrix were rectangular (m x n) instead of square?
  • If the matrix represents a large social network graph, why might computing the principal eigenvalue (the one with the largest absolute value) be useful for ranking nodes?
  • How does the concept of eigenvalues extend to higher-order tensors in machine learning, such as in Principal Component Analysis (PCA) or spectral clustering?