1. Choose a $2\times2$ matrix
Edit $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$. The page computes its singular values from $A^TA$.
2. Geometry: unit circle to ellipse
Blue: unit circle. Orange: image under $A$. The ellipse semi-axis lengths are the singular values.
3. Rank-$k$ approximation
For a $2\times2$ matrix, the rank-one truncation is $A_1=\sigma_1u_1v_1^T$.
4. Pseudoinverse least squares
Solve $Ax=b$ in least-squares sense for $b=(1,2)^T$ using $A^+$.
5. PCA demo
Move the covariance parameters to rotate and stretch a synthetic data cloud. The red direction is the first principal component.
6. PCA geometry
7. Independent-study practice tasks with answers
Task A. Why are singular values always nonnegative?
Answer: They are square roots of eigenvalues of $A^TA$, and $A^TA$ is positive semidefinite.
Answer: They are square roots of eigenvalues of $A^TA$, and $A^TA$ is positive semidefinite.
Task B. If singular values are $8,2,0$, what are the rank and spectral norm?
Answer: Rank is $2$ and $\|A\|_2=8$.
Answer: Rank is $2$ and $\|A\|_2=8$.
Task C. If singular values are $10,4,1$, what is the best rank-one spectral norm error?
Answer: $\|A-A_1\|_2=\sigma_2=4$.
Answer: $\|A-A_1\|_2=\sigma_2=4$.
Similar practice. If singular values are $12,5,2,0.5$, what is the best rank-two spectral norm error?
Answer: $\sigma_3=2$.
Answer: $\sigma_3=2$.