Transformations: Eigenvalues, Diagonalization, and Quadratic Forms

Vocabulary flashcards covering core concepts of linear transformations, eigenvalues, eigenvectors, Cayley-Hamilton theorem, matrix diagonalization, and quadratic form reductions.

Eigenvalue

A scalar ̱\lambda for a square matrix $A$ of order $n$ for which there exists a non-zero vector $X$ such that $AX = \lambda X$.

Eigenvector

A non-zero vector $X$ corresponding to an eigenvalue $\lambda$ that satisfies the linear equation $AX = \lambda X$ for a square matrix $A$.

Characteristic Equation

The equation $\det(A - \lambda I) = 0$ (or $|A - \lambda I| = 0$) whose roots give the eigenvalues of a square matrix $A$.

Principal Directions

The special directions defined by vectors $x$ for which a linear deformation $y = Ax$ stretches or contracts an elastic membrane without altering its direction (i.e., $Ax$ is parallel to $x$).

Principal Stretches

The scaling factors $\lambda$ corresponding to the principal directions during the linear deformation $y = Ax$ of a membrane.

Singular Deformation

A linear deformation of an elastic membrane where one eigenvalue is zero ($\lambda_2 = 0$), causing the boundary circle to completely collapse into a line segment rather than an ellipse.

Trace of a Matrix

The sum of the main diagonal elements of a square matrix $A$, which is equal to the sum of all its eigenvalues.

Determinant of a Matrix

The scalar value $\det(A)$ of a square matrix $A$, which equals the product of all its eigenvalues.

Cayley-Hamilton Theorem

The theorem stating that every square matrix $A$ satisfies its own characteristic equation.

Similar Matrices

Two square matrices $A$ and $B$ of order $n \times n$ for which there exists an invertible matrix $P$ such that $B = P^{-1}AP$.

Modal Matrix

An invertible change-of-basis matrix $P$ whose columns consist of the eigenvectors of a matrix $A$.

Similarity Transformation

The transformation of a square matrix $A$ into a diagonal matrix $D$ using an invertible modal matrix $P$ such that $P^{-1}AP = D$.

Orthogonal Transformation

A special similarity transformation where the change-of-basis matrix $P$ is an orthogonal matrix ($P^{-1} = P^T$), yielding $P^T A P = D$.

Normalized Modal Matrix

A matrix $N$ whose columns are the normalized (unit length) mutually orthogonal eigenvectors of a matrix $A$.

Orthogonally Diagonalizable Matrix

A real symmetric matrix $A$ for which there exists an orthogonal matrix $N$ ($N^T N = I$) and a diagonal matrix $D$ such that $D = N^T A N$.

Quadratic Form

A homogeneous polynomial of the second degree in any number of variables, expressible in matrix notation as $Q = X^T A X$ where $A$ is a real symmetric matrix.

Canonical Form

A simplified form of a quadratic form that contains only square terms of variables and no cross-product terms, written as $\sum_{i=1}^n \lambda_i y_i^2$.

Rank of a Quadratic Form

The number of non-zero coefficients (or non-zero eigenvalues) in the canonical form of a quadratic form, denoted by $r$.

Index of a Quadratic Form

The number of positive coefficients (or positive eigenvalues) in the canonical form of a quadratic form, denoted by $p$.

Signature of a Quadratic Form

The difference between the number of positive and negative coefficients in the canonical form, calculated as $s = 2p - r$.

Positive Definite Quadratic Form

A quadratic form $Q = X^T A X$ whose canonical form has all strictly positive eigenvalues ($\lambda_i > 0$ for all $i$).

Positive Semidefinite Quadratic Form

A quadratic form $Q = X^T A X$ whose canonical form has non-negative eigenvalues ($\lambda_i \ge 0$) with at least one eigenvalue equal to $0$.

Negative Definite Quadratic Form

A quadratic form $Q = X^T A X$ whose canonical form has all strictly negative eigenvalues ($\lambda_i < 0$ for all $i$).

Negative Semidefinite Quadratic Form

A quadratic form $Q = X^T A X$ whose canonical form has non-positive eigenvalues ($\lambda_i \le 0$) with at least one eigenvalue equal to $0$.

Indefinite Quadratic Form

A quadratic form $Q = X^T A X$ whose matrix $A$ possesses both positive and negative eigenvalues.

Principal Minors (Principal Sub-determinants)

The determinants $D_1, D_2, \dots, D_n$ of the upper-left square submatrices of a symmetric matrix $A$, used to evaluate definiteness without explicit canonical reduction.