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5.1.2-matrices.md

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5.1.2 Matrices

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  1. [E] Why do we say that matrices are linear transformations?

  2. [E] What’s the inverse of a matrix? Do all matrices have an inverse? Is the inverse of a matrix always unique?

  3. [E] What does the determinant of a matrix represent?

  4. [E] What happens to the determinant of a matrix if we multiply one of its rows by a scalar $$t \times R$$?

  5. [M] A $$4 \times 4$$ matrix has four eigenvalues $$3, 3, 2, -1$$. What can we say about the trace and the determinant of this matrix?

  6. [M] Given the following matrix:
    $$ \begin{bmatrix} 1 & 4 & -2 \ -1 & 3 & 2 \ 3 & 5 & -6 \end{bmatrix} $$

    Without explicitly using the equation for calculating determinants, what can we say about this matrix’s determinant?

    Hint: rely on a property of this matrix to determine its determinant.

  7. [M] What’s the difference between the covariance matrix $$A^TA$$ and the Gram matrix $$AA^T$$?

  8. Given $$A \in R^{n \times m}$$ and $$b \in R^n$$

    1. [M] Find $$x$$ such that: $$Ax = b$$.
    2. [E] When does this have a unique solution?
    3. [M] Why is it when A has more columns than rows, $$Ax = b$$ has multiple solutions?
    4. [M] Given a matrix A with no inverse. How would you solve the equation $$Ax = b$$? What is the pseudoinverse and how to calculate it?

  9. Derivative is the backbone of gradient descent.

    1. [E] What does derivative represent?
    2. [M] What’s the difference between derivative, gradient, and Jacobian?

  10. [H] Say we have the weights $$w \in R^{d \times m}$$ and a mini-batch $$x$$ of $$n$$ elements, each element is of the shape $$1 \times d$$ so that $$x \in R^{n \times d}$$. We have the output $$y = f(x; w) = xw$$. What’s the dimension of the Jacobian $$\frac{\delta y}{\delta x}$$?

  11. [H] Given a very large symmetric matrix A that doesn’t fit in memory, say $$A \in R^{1M \times 1M}$$ and a function $$f$$ that can quickly compute $$f(x) = Ax$$ for $$x \in R^{1M}$$. Find the unit vector $$x$$ so that $$x^TAx$$ is minimal.

    Hint: Can you frame it as an optimization problem and use gradient descent to find an approximate solution?