• Title: Positive Definite Matrices

  • Series: Abstract Linear Algebra

  • Chapter: General inner products

  • YouTube-Title: Abstract Linear Algebra 11 | Positive Definite Matrices

  • Bright video: https://youtu.be/ZuOrdYMkQmU

  • Dark video: https://youtu.be/e5Q8VGC4Zwc

  • Quiz: Test your knowledge

  • PDF: Download PDF version of the bright video

  • Dark-PDF: Download PDF version of the dark video

  • Print-PDF: Download printable PDF version

  • Thumbnail (bright): Download PNG

  • Thumbnail (dark): Download PNG

  • Subtitle on GitHub: ala11_sub_eng.srt missing

  • Timestamps (n/a)
  • Subtitle in English (n/a)
  • Quiz Content

    Q1: Which matrix $A \in \mathbb{C}^{2 \times 2}$ is clearly not positive definite?

    A1: $ A = \begin{pmatrix} 1 & 1 \ 0 & 1 \end{pmatrix} $

    A2: $ A = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} $

    A3: $ A = \begin{pmatrix} 5 & 1 \ 1 & 1 \end{pmatrix} $

    A4: $ A = \begin{pmatrix} 2 & 3 \ 3 & 5 \end{pmatrix} $

    Q2: What is not equivalent for $A = \begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{pmatrix} \in \mathbb{C}^{2 \times 2}$ being a positive-definite matrix?

    A1: $a_{11} > 0$ and $ \det(A) > 0$.

    A2: $A$ is selfadjoint and $\langle x, A x \rangle_{standard} > 0 $ for all $x \in \mathbb{C}^2 \setminus { 0 } $.

    A3: $A$ is selfadjoint and all eigenvalues are positive.

    A4: $\langle x, A x \rangle_{\text{standard}} > 0 $ for all $x \in \mathbb{C}^2 \setminus { 0 } $.

    Q3: Apply a Gaussian step to the matrix $A = \begin{pmatrix} 2 & 1 \ 1 & 1 \end{pmatrix}$ that is allowed according to the video such that one can check if $A$ is positive definite. Which matrix do we get?

    A1: $A = \begin{pmatrix} 2 & 1 \ 0 & \frac{1}{2} \end{pmatrix}$

    A2: $A = \begin{pmatrix} - 1 & -\frac{1}{2} \ 1 & 1 \end{pmatrix}$

    A3: $A = \begin{pmatrix} 1 & 1 \ 2 & 1 \end{pmatrix}$

    A4: $A = \begin{pmatrix} 2 & 1 \ -2 & -2 \end{pmatrix}$

  • Last update: 2024-10

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