How can a matrix be expressed as a sum of rank-one terms using its SVD?

For rank $r$, $M=\sum_{i=1}^{r}\sigma_i u_i v_i^*$, where each term is a weighted outer product.

How are singular values related to $M^*M$?

The nonzero singular values of $M$ are the non-negative square roots of the nonzero eigenvalues of $M^*M$.

How is the pseudoinverse constructed from an SVD?

For $M=U\Sigma V^*$, $M^+=V\Sigma^+U^*$; nonzero singular values are reciprocated and zero values stay zero.

Which SVD vectors span the null space of a matrix?

Right-singular vectors associated with zero singular values span the null space of $M$.

What does numerical rank estimate?

It estimates rank by treating singular values below a scale- and precision-informed tolerance as zero.

Linear algebraLesson 1 of 2 in the course Eigenvalues and eigenvectors

Singular Value Decomposition: Geometry, Rank, and Applications

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Summary

For an $m \times n$ complex matrix $M$ , the SVD has the form

Here, $U$ is an $m \times m$ unitary matrix, $V$ is an $n \times n$ unitary matrix, and $V^*$ is the conjugate transpose of $V$ . The matrix $\Sigma$ is an $m \times n$ rectangular diagonal matrix: its only potentially nonzero entries lie on its main diagonal, and those entries are non-negative real numbers.

All flashcards

How can a matrix be expressed as a sum of rank-one terms using its SVD?: For rank $r$ , $M=\sum_{i=1}^{r}\sigma_i u_i v_i^*$ , where each term is a weighted outer product.
How are singular values related to $M^*M$ ?: The nonzero singular values of $M$ are the non-negative square roots of the nonzero eigenvalues of $M^*M$ .
How is the pseudoinverse constructed from an SVD?: For $M=U\Sigma V^*$ , $M^+=V\Sigma^+U^*$ ; nonzero singular values are reciprocated and zero values stay zero.
Which SVD vectors span the null space of a matrix?: Right-singular vectors associated with zero singular values span the null space of $M$ .
What does numerical rank estimate?: It estimates rank by treating singular values below a scale- and precision-informed tolerance as zero.

Quiz questions

For an $m\times n$ matrix, what are the dimensions of $U$ and $V$ in the full SVD?Answer: $U$ is $m\times m$ and $V$ is $n\times n$ .
Which statement describes the scope of the singular value decomposition?Answer: It exists for every rectangular or square matrix.
What are the nonzero eigenvalues of $M^*M$ in terms of the singular values of $M$ ?Answer: They are the squares of the nonzero singular values.
What does the Eckart–Young theorem say about truncating an SVD?Answer: The truncated SVD gives a best rank- $t$ approximation in the Frobenius norm.
For a full-rank matrix, what is the usual SVD expression for its condition number?Answer: $\kappa=\sigma_{\max}/\sigma_{\min}$ .

Based on the Wikipedia article Singular value decomposition, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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