Editorial Staff
Let’s consider a square matrix $A$ of size $n \times n$.
Step 1: Determinant of $(A – \lambda I)$
Start by subtracting the scalar multiple of the identity matrix $(\lambda I)$ from matrix $A$, where $\lambda$ is an unknown eigenvalue.
\[ B = A – \lambda I \]
In this step, we create a new matrix $B$ by subtracting $\lambda$ times the identity matrix $I$ from matrix $A$. The identity matrix $I$ has the same size as matrix $A$ and contains ones on the diagonal and zeros elsewhere.
Step 2: Calculate Determinant of $B$
Calculate the determinant of matrix $B$ using any suitable method, such as cofactor expansion or row reduction.
\[ \det(B) = \begin{vmatrix} a_{11} – \lambda & a_{12} & \ldots & a_{1n} \\ a_{21} & a_{22} – \lambda & \ldots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \ldots & a_{nn} – \lambda \end{vmatrix} \]
Related | How to Calculate Determinant?
In this step, we calculate the determinant of matrix $B$. The determinant is a scalar value that represents certain properties of the matrix. In LaTeX, we use the “vmatrix” environment to represent the determinant.
Step 3: Set Determinant to Zero
Set the determinant of $B$ equal to zero and solve for $\lambda$.
\[ \det(B) = 0 \]
In this step, we set the determinant of matrix $B$ equal to zero. This is because the eigenvalues are the values of $\lambda$ that satisfy this equation.
Step 4: Solve for Eigenvalues
Solve the equation obtained in the previous step to find the values of $\lambda$.
These values are the eigenvalues of matrix $A$.
