What actually are eigenvalues and eigenvectors?
For any square matrix $A$, an Eigenvector ($v$) and Eigenvalue ($\lambda$) satisfy the linear algebra equation. In other words, when $A$ is multiplied by the eigenvector ($v$), the resulting vector points in the exact same direction (or opposite), scaled only by the scaler factor $\lambda$.
How to find eigenvalue?
To find eigenvalue, we can formulate equation as following:
Example:
Now, to calculate eigenvectors:
Similarly, we can calcuate for $\lambda$ = -1.
References:
[1] P. G. Strang, “Eigenvalues and Eigenvectors: Introduction to Eigenvalues,” 2016, https://math.mit.edu/~gs/linearalgebra/ila5/linearalgebra5_6-1.pdf.
[2] Dan Margalit and Joseph Rabinoff, “Eigenvalues and Eigenvectors,” in Interactive Linear Algebra, [Online]. Available: https://textbooks.math.gatech.edu/ila/eigenvectors.html
[3] “M10 Eigenvalues and Eigenvectors,” July 15, 2021, https://learninglab.rmit.edu.au/sites/default/files/M10_Eigenvalues_and_eigenvectors.pdf.
[4] Dan Margalit and Joseph Rabinoff, “Eigenvalues and Eigenvectors,” in Interactive Linear Algebra, [Online]. Available: https://textbooks.math.gatech.edu/ila/eigenvectors.html
[5] R. Pierce, “Eigenvector and Eigenvalue,” Math is Fun, 2026. https://www.mathsisfun.com/algebra/eigenvalue.html