# Properties of Eigenvalues

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
• Basis of eigenvectors results in a diagonal matrix for $T$
• Every linear map has an upper triangular matrix representation

## Eigenvalues, nullity and invertibility

$T:V\to V$

Zero is an eigenvalue iff dim null $T$ > 0.

$v$: eigenvector with eigenvalue 0 $\Leftrightarrow$ $Tv=0$, $v\ne0$

$T$ is invertible iff no eigenvalue is zero.

Corollary of above

$T$: invertible with eigenvalue $\lambda\ne0$, eigenvector $v$. Then, $1/\lambda$ is an eigenvalue of $T^{-1}$ with eigenvector $v$.

$Tv=\lambda v$ implies $T^{-1}v=(1/\lambda)v$

## Eigenvalues and transpose

Suppose $\lambda$ is an eigenvalue of $A$. Then, $\lambda$ is an eigenvalue of $A^T$.

Proof

$\lambda$: $A-\lambda I$ is non-invertible

rank$(A-\lambda I)=$ rank$(A^T-\lambda I)$

So, $A^T-\lambda I$ is non-invertible

## Eigenvalues and determinant

Determinant is equal to the product of eigenvalues.

Proof

• Find upper triangular matrix representation for $T$.

• Determinant is product of diagonal entries, which are eigenvalues.

## Trace of an operator

$T:V\to V$, $A$: matrix w.r.t. some basis

Trace of $T$, denoted tr$(T)$, is defined as sum of diagonal elements of $A$

why is definition valid?

Sum of diagonal elements of $SAS^{-1}$ and $A$ are equal.

Proof

• tr$(AB) =$ tr$(BA)$ (exercise)

• tr$(SAS^{-1})=$ tr$(S(AS^{-1}))=$ tr$((AS^{-1})S)=$ tr$(A)$

## Eigenvalues and trace

Trace is equal to the sum of eigenvalues.

Proof

• Find upper triangular matrix representation for $T$.

• Trace is sum of diagonal entries, which are eigenvalues.