2.4 Eigenvalues and Eigenvectors

• What is an eigenvector of a matrix?

  Non-zero scaled vector
• The scalar used for scaling an eigenvector is called the eigenvalue
• Mathematically, for a matrix $A$, a vector $v$ is an eigenvector with eigenvalue $\lambda$ if $Av =$$\lambda v$.
• Give an example of an eigenvector and its corresponding eigenvalue for the matrix $A =$$\begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}$.


  $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ with eigenvalue 2
• An eigenvector is always a non-zero vector.
• An eigenvector remains an eigenvector if it is multiplied by a non-zero scalar.
• What is the key characteristic of an eigenvector-eigenvalue pair for a matrix?

  Av = \lambda v</latex>
• Match the term with its definition:

  Eigenvector ↔️ Non-zero vector scaled by eigenvalue

  Eigenvalue ↔️ Scalar used to scale eigenvector
• What is an eigenvector of a matrix?

  Non-zero vector scaled by matrix
• Mathematically, for a matrix $A$, a vector $v$ is an eigenvector and $\lambda$ is its corresponding eigenvalue if $Av =$$\lambda v$, which means the scaled version of the vector is the result of multiplying the matrix by the eigenvector
• For the matrix $A =$$\begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}$, what is the eigenvalue corresponding to the eigenvector $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$?


  2
• Match the concepts with their definitions:

  1️⃣ Eigenvector

  2️⃣ Non-zero vector scaled by matrix

  3️⃣ Eigenvalue

  4️⃣ Scalar used for scaling
• For a matrix $A$, a vector $v$ is an eigenvector if $Av =$$\lambda v$.
• An eigenvector is a non-zero vector that, when multiplied by a matrix, remains in the same direction
• What is the role of the eigenvalue in matrix transformations?

  Amount of scaling applied
• The eigenvector $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ corresponds to the eigenvalue 2 for the matrix $A =$$\begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}$.
• The characteristic equation is derived from the eigenvector/eigenvalue equation Av = \lambda v</latex> by rewriting it as $(A - \lambda I)v =$$0$, where $I$ is the identity matrix.
• What is the characteristic equation for a matrix $A$?


  $\det(A - \lambda I) =$$0$
• For a matrix $A$, the determinant of $(A - \lambda I)$ must be zero for $\lambda$ to be an eigenvalue.
• What is the eigenvector/eigenvalue equation from which the characteristic equation is derived?

  $Av =$$\lambda v$
• The characteristic equation is derived from the condition $\det(A - \lambda I) =$$0$ for a matrix $A$.
• Steps to calculate eigenvalues using the characteristic equation

  1️⃣ Formulate $(A - \lambda I)$

  2️⃣ Find the determinant of $(A - \lambda I)$

  3️⃣ Set the determinant equal to zero and solve for $\lambda$
• What is the definition of an eigenvector?

  Non-zero vector scaled by matrix
• For a matrix $A$, a vector $v$ is an eigenvector if $Av =$$\lambda$v
• Match the concept with its role in linear algebra:

  Eigenvector ↔️ Direction unchanged by matrix transformation

  Eigenvalue ↔️ Amount of scaling applied

  Characteristic equation ↔️ Condition to find eigenvalues
• What is the first step to calculate eigenvalues using the characteristic equation?

  Form $A - \lambda I$
• To find eigenvectors corresponding to each eigenvalue, use the equation $Av =$$\lambda$v
• Normalizing an eigenvector ensures its magnitude is equal to 1.
• Steps to normalize an eigenvector

  1️⃣ Calculate the magnitude of the eigenvector

  2️⃣ Divide each component of the eigenvector by its magnitude
• What does normalizing an eigenvector do to its magnitude?

  Makes it equal to 1
• Steps to normalize an eigenvector

  1️⃣ Calculate the magnitude

  2️⃣ Divide each component by the magnitude
• Normalizing an eigenvector simplifies calculations and ensures consistent representation.
• Normalize the eigenvector v = \begin{pmatrix} 3 \\ 4 \end{pmatrix}</latex>.

  $\hat{v} =$$\begin{pmatrix} \frac{3}{5} \\ \frac{4}{5} \end{pmatrix}$
• An eigenvector of a matrix is a non-zero vector that, when multiplied by the matrix, results in a scaled version of the same vector
• What is the scalar used for scaling an eigenvector called?

  Eigenvalue
• For matrix $A$, vector $v$ is an eigenvector with eigenvalue $\lambda$ if $Av =$$\lambda v$.
• Give an example of an eigenvector and its corresponding eigenvalue for A = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}</latex>.

  $v =$$\begin{pmatrix} 1 \\ 0 \end{pmatrix}$, $\lambda =$$2$
• The characteristic equation is derived from the eigenvalue equation $Av =$$\lambda v$ and transformed to $(A - \lambda I)v =$$0$, where $I$ is the identity matrix
• What is the condition for the characteristic equation to be satisfied?

  $\det(A - \lambda I) =$$0$
• The characteristic equation $\det(A - \lambda I) =$$0$ is used to find the eigenvalues of a matrix.