Lecture 15: Matrices A(t) Depending on t, Derivative = dA/dt
Description
This lecture is about changes in eigenvalues and changes in singular values. When matrices move, their inverses, their eigenvalues, and their singular values change. Professor Strang explores the resulting formulas.
Summary
Matrices \(A(t)\) depending on \(t / \)Derivative \(= dA/dt\)
The eigenvalues have derivative \(y(dA/dt)x\).\(x\) = eigenvector, \(y\) = eigenvector of transpose of \(A\)
Eigenvalues from adding rank-one matrix are interlaced.
Related section in textbook: III.1-2
Instructor: Prof. Gilbert Strang
Problems for Lecture 15
From textbook Sections III.1 - III.2
1. A unit vector \(\boldsymbol{u}(t)\) describes a point moving around on the unit sphere \(\boldsymbol{u}^{\mathtt{T}} \boldsymbol{u} =1\). Show that the velocity vector \(d\boldsymbol{u}/dt\) is orthogonal to the position: \(\boldsymbol{u}^{\mathtt{T}}(d\boldsymbol{u}/dt)=0\).
2. Suppose you add a positive semidefinite rank two matrix to \(S\). What interlacing inequalities will connect the eigenvalues \(\lambda\) of \(S\) and \(\alpha\) of \(S+\boldsymbol{uu}^{\mathtt{T}}+\boldsymbol{vv}^{\mathtt{T}}\) ?
5. Find the eigenvalues of \(A_3\) and \(A_2\) and \(A_1\). Show that they are interlacing:
$$A_3 = \left[\begin{matrix}1 & -1&0 \\ -1 &2&-1\\ 0 &-1&1\end{matrix}\right] \hspace{12pt} A_2 = \left[\begin{matrix}1 & -1\\ -1 & 2\end{matrix}\right] \hspace{12pt} A_1 = \left[\begin{matrix}1\end{matrix}\right] $$
