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Learning objectives

What you will be able to explain

Basic inverse algebra
Recognize the Woodbury identity
Sherman-Morrison scalar case
Implications involving inverses
Choose a convergent inverse expansion
Generalized inverse conditions

Section 01

Basic inverse algebra to Sherman-Morrison scalar case

Basic inverse algebra

Inverse distributes over products only with reversed order, not over sums.

Guided checkpoint

Assume all needed inverses exist.

Source: Sec. 3.1, p. 17

Recognize the Woodbury identity

The matrix inversion lemma turns a low-rank update into a smaller central inverse.

Guided checkpoint

Which form inverts $A+UCV$ using an inverse in the smaller update space?

Source: Sec. 3.2, pp. 18-19

Sherman-Morrison scalar case

Sherman-Morrison gives

I-uv^T/(1+v^Tu)

; the entry is

1-2/3=1/3

Guided checkpoint

Let $A=I_2$ , $u=[1,0]^T$ , $v=[2,0]^T$ . Compute the top-left entry of $(A+uv^T)^{-1}$ .

Source: Sec. 3.2, pp. 18-19

Section 02

Implications involving inverses to Generalized inverse conditions

Implications involving inverses

Combine definitions with

A^2=A

: multiplying by

A^{-1}

yields

A=I

Guided checkpoint

Audit each statement.

Source: Sec. 3.3, p. 20

Choose a convergent inverse expansion

The Neumann series is the matrix analogue of the geometric series and requires a convergence condition.

Guided checkpoint

When the spectral radius of $A$ is below one, which series equals $(I-A)^{-1}$ ?

Source: Sec. 3.4, p. 20

Generalized inverse conditions

The four Penrose equations uniquely characterize the pseudoinverse.

Guided checkpoint

Match each Moore-Penrose condition.

Source: Secs. 3.5-3.6, pp. 21-23

Section 03

Pseudoinverse consequences

Every matrix has a Moore-Penrose pseudoinverse; the associated products are orthogonal projectors.

Guided checkpoint

Let $A=U Sigma V^T$ be an SVD.

Source: Sec. 3.6, pp. 21-23

Knowledge check

Turn understanding into recall.

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