More on Eigenvalues, Eigenvectors, Diagonalization

Andrew Thangaraj

Aug-Nov 2020

Recap

  • Vector space \(V\) over a scalar field \(F\)
    • \(F\): real field \(\mathbb{R}\) or complex field \(\mathbb{C}\) in this course
  • \(m\times n\) matrix A represents a linear map \(T:F^n\to F^m\)
    • dim null \(T\) + dim range \(T\) = dim \(V\)
  • Linear equation: \(Ax=b\)
    • Solution (if it exists): \(u+\) null\((A)\)
  • Four fundamental subspaces of a matrix
    • Column space, row space, null space, left null space
  • Eigenvalue \(\lambda\) and Eigenvector \(v\): \(Tv=\lambda v\)
    • Distinct eigenvalues have independent eigenvectors

Repeatedly applying an operator on a vector

\(T:V\to V\), operator and \(v\in V\), \(v\ne 0\), \(F=\mathbb{C}\) and dim \(V=n\)

\(T\) can be applied repeatedly to \(v\)

\(v\), \(Tv\), , \(T^nv\): \(n+1\) vectors

Linearly dependent in \(V\)

\(a_0v+a_1Tv+\cdots+a_nT^nv=0\), \(a_i\in \mathbb{C}\)

Let \(m=\) max \(i\) s.t. \(a_i\ne 0\), \(m\ge 1\)

\((a_0+a_1T+\cdots+a_mT^m)v=0\), \(v\ne0\), \(a_m\ne 0\)

Let \(a_0+a_1x+\cdots+a_mx^m=a_m(x-\lambda_1)\cdots(x-\lambda_m)\)

Operator algebra - they can be added, multiplied

\(a_0+a_1T+\cdots+a_mT^m=a_m(T-\lambda_1I)\cdots(T-\lambda_mI)\)

\(a_m(T-\lambda_1I)\cdots(T-\lambda_mI)v=0\), \(v\ne 0\)

Existence of eigenvalues: another proof

Since \((T-\lambda_1I)\cdots(T-\lambda_mI)v=0\), \(v\ne 0\), there exist at least one \(i\) s.t. \(T-\lambda_iI\) is non-invertible

Proof

Contradiction: if \(T-\lambda_iI\) is invertible for every \(i\), then \(v=0\)

\(T-\lambda_iI\): non-invertible implies \(\lambda_i\) is an eigenvalue

This proves existence of one eigenvalue for an operator (\(F=\mathbb{C}\))

There are at most dim \(V\) distinct eigenvalues for an operator

Proof

Eigenvectors of distinct eigenvalues are linearly independent

Eigenvalues of diagonal and triangular matrices

Diagonal elements of a diagonal matrix are eigenvalues of the corresponding operator

Proof

\(T-\lambda I\): non-invertible if \(\lambda=\) a diagonal element

Diagonal elements of a triangular matrix are eigenvalues of the corresponding operator

Proof

\(T-\lambda I\): non-invertible if \(\lambda=\) a diagonal element

Diagonalization

If \(T\) has \(n=\) dim \(V\) distinct eigenvalues, the eigenvectors form a basis for \(V\)

Proof

Eigenvectors are independent for distinct eigenvalues

\(n\) linearly independent eigenvectors form a basis

In the above basis of eigenvectors, \(T\) is diagonal

Proof

\(v_i\): eigenvector, \(Tv_i=\lambda_i v_i\), \(i=1,\ldots,n\)

\(\{v_1,\ldots,v_n\}\): basis

\(Tv_i\): has 1 in \(i\)-th coordinate, zero elsewhere

Colab notebook for eigenvalues and eigenvectors

Examples: Repeated Eigenvalues

\(\begin{bmatrix} 1&0&0\\ 0&1&0\\ 0&0&1 \end{bmatrix}\)

Eigenvalues: \(1, 1, 1\)

Eigenvectors: \((1,0,0)\), \((0,1,0)\), \((0,0,1)\)

\(\begin{bmatrix} 1&2&0\\ 0&1&0\\ 0&0&1 \end{bmatrix}\)

Eigenvalues: \(1, 1, 1\)

Eigenvectors: \((1,0,0)\), \((0,0,1)\)

\(\begin{bmatrix} 1&2&0\\ 0&1&3\\ 0&0&1 \end{bmatrix}\)

Eigenvalues: \(1, 1, 1\)

Eigenvectors: \((1,0,0)\)

Towards “simple” matrices

Eigenvalues are useful in obtaining simple matrices

Basis of eigenvectors results in a diagonal matrix

When eigenvalues are repeated, there may not be enough eigenvectors

What can be said, in general, about the “simplest” matrix for an operator?