Comprehensive Study Guide on Unit 1: Matrices

Characteristic Equation

For a square matrix $A$, the equation $\|A - \lambda I\| = 0$ is called the characteristic equation of matrix $A$, where $I$ is the identity matrix of the same order as $A$ and $\lambda$ is a scalar parameter.

Characteristic Roots / Eigenvalues

The roots obtained by solving the characteristic equation $\|A - \lambda I\| = 0$ for an $n \times n$ matrix $A$.

Eigenvector / Latent Vector

A non-zero solution vector $X$ satisfying the matrix equation $AX = \lambda X$ (or equivalently $(A - \lambda I)X = 0$) corresponding to an eigenvalue $\lambda$.

Characteristic Polynomial

The polynomial in $\lambda$ obtained by expanding the determinant $\|A - \lambda I\|$ without setting it equal to zero.

Cayley-Hamilton Theorem

States that every square matrix satisfies its own characteristic equation. If the characteristic equation is $\lambda^n + c_{n-1}\lambda^{n-1} + \dots + c_0 = 0$, then $A^n + c_{n-1}A^{n-1} + \dots + c_0 I = 0$.

Algebraic Multiplicity (AM)

The multiplicity of an eigenvalue as a root of the characteristic polynomial.

Geometric Multiplicity (GM)

The maximum number of linearly independent eigenvectors corresponding to a specific eigenvalue $\lambda$.

Trace Property of Eigenvalues

The sum of the eigenvalues of a matrix $A$ equals the trace (sum of main diagonal elements) of matrix $A$: $\sum_{i=1}^n \lambda_i = \text{Trace}(A)$

Determinant Property of Eigenvalues

The product of the eigenvalues of a matrix $A$ equals the determinant of matrix $A$: $\prod_{i=1}^n \lambda_i = \|A\|$

Inverse Eigenvalue Property

If $\lambda_1, \lambda_2, \dots, \lambda_n$ are eigenvalues of an invertible matrix $A$, then the eigenvalues of $A^{-1}$ are $\frac{1}{\lambda_1}, \frac{1}{\lambda_2}, \dots, \frac{1}{\lambda_n}$.

Orthogonal Matrix

A real square matrix $A$ satisfying $A A^T = A^T A = I$, or equivalently $A^T = A^{-1}$.

Quadratic Form

A homogeneous polynomial of the second degree in any number of variables.

Rank of a Quadratic Form ($r$)

The total number of non-zero square terms present in the canonical form of a quadratic form.

Index of a Quadratic Form ($s$)

The number of positive square terms present in the canonical form of a quadratic form.

Signature of a Quadratic Form

The difference between the number of positive and negative square terms in canonical form: $\text{Signature} = 2s - r$.

Positive Definite Quadratic Form

A quadratic form in which all eigenvalues of its corresponding symmetric matrix are strictly positive ($> 0$).