mth215 exam 2 terms

Matrix Multiplication Transpose Rule

(AB)^T = B^T A^T. The transpose of a product equals the product of the transposes in reverse order.

Inverse of a 2x2 Matrix Formula

For A = [[a, b], [c, d]], if det(A) = ad - bc != 0, then A^-1 = (1/(ad - bc)) * [[d, -b], [-c, a]].

Inverse of a Matrix Product Rules

(AB)^-1 = B^-1 A^-1. The inverse of a product equals the product of the inverses in reverse order.

Algorithm for Finding Matrix Inverse

Form the augmented matrix [A | I_n] and row reduce to [I_n | A^-1]. If A cannot be reduced to I_n, A is not invertible.

Invertible Matrix Theorem (Linear Independence)

An n x n matrix A is invertible if and only if its columns form a linearly independent set in R^n.

Invertible Matrix Theorem (Span)

An n x n matrix A is invertible if and only if its columns span R^n.

Invertible Matrix Theorem (Trivial Solution)

An n x n matrix A is invertible if and only if the equation A x = 0 has only the trivial solution x = 0.

Invertible Matrix Theorem (Determinant)

An n x n matrix A is invertible if and only if det(A) != 0.

Invertible Matrix Theorem (Eigenvalues)

An n x n matrix A is invertible if and only if 0 is NOT an eigenvalue of A.

Determinant of a Triangular Matrix

The determinant of a triangular (upper or lower) or diagonal matrix is the product of the entries on its main diagonal.

Row Replacement Effect on Determinant

Adding a scalar multiple of one row to another row leaves det(A) unchanged: det(B) = det(A).

Row Interchange Effect on Determinant

Interchanging two rows of a matrix multiplies its determinant by -1: det(B) = -det(A).

Row Scaling Effect on Determinant

Multiplying one row of an n x n matrix by scalar k multiplies its determinant by k: det(B) = k * det(A).

Determinant of Scalar Multiple of Matrix

For an n x n matrix A and scalar k, det(k A) = (k^n) * det(A).

Multiplicative Property of Determinants

For any two n x n matrices A and B, det(A B) = det(A) * det(B).

Determinant of Transpose

For any square matrix A, det(A^T) = det(A).

Determinant of Matrix Inverse

If A is invertible, det(A^-1) = 1 / det(A).

Eigenvector Definition

A nonzero vector x such that A x = lambda x for some scalar lambda.

Eigenvalue Definition

A scalar lambda such that A x = lambda x has a nontrivial solution x != 0.

Eigenspace Definition

The null space of (A - lambda I_n), consisting of all eigenvectors corresponding to lambda plus the zero vector.

Basis for an Eigenspace Algorithm

Find the general solution to (A - lambda I_n) x = 0 by row reducing [A - lambda I_n | 0] and expressing the solution in parametric vector form.

Linear Independence of Eigenvectors

Eigenvectors corresponding to distinct eigenvalues are linearly independent.

Characteristic Equation

det(A - lambda I_n) = 0. The roots of this polynomial equation are the eigenvalues of A.

Algebraic Multiplicity of an Eigenvalue

The multiplicity of an eigenvalue lambda as a root of the characteristic polynomial det(A - lambda I_n) = 0.

Geometric Multiplicity of an Eigenvalue

The dimension of the eigenspace Nul(A - lambda I_n), which equals the maximum number of linearly independent eigenvectors for lambda.

Matrix Similarity Definition

An n x n matrix A is similar to B if there exists an invertible matrix P such that A = P B P^-1.

Eigenvalues of Similar Matrices

If A and B are similar n x n matrices, they have the exact same characteristic polynomial and therefore the same eigenvalues.

Matrix Diagonalization Definition

An n x n matrix A is diagonalizable if A = P D P^-1, where D is a diagonal matrix and P is an invertible matrix.

The Diagonalization Theorem

An n x n matrix A is diagonalizable if and only if A has n linearly independent eigenvectors.

Constructing P and D for Diagonalization

P is built using n linearly independent eigenvectors of A as its columns, and D is a diagonal matrix whose diagonal entries are the corresponding eigenvalues in the same column order.

Computing Matrix Powers via Diagonalization

If A = P D P^-1, then A^k = P (D^k) P^-1, where D^k is computed by raising each diagonal entry of D to the k-th power.

Inner Product (Dot Product) Formula

The inner product of u and v in R^n is u . v = u^T v = u_1 v_1 + u_2 v_2 + … + u_n v_n.

Length (Norm) of a Vector

The norm ||v|| of vector v is sqrt(v . v) = sqrt(v_1^2 + v_2^2 + … + v_n^2).

Orthogonal Vectors Definition

Two vectors u and v in R^n are orthogonal if and only if u . v = 0.

Pythagorean Theorem for Orthogonal Vectors

Two vectors u and v are orthogonal if and only if ||u + v||^2 = ||u||^2 + ||v||^2.

Orthogonal Complement Definition

The set W^perp of all vectors in R^n that are orthogonal to every vector in subspace W.

Fundamental Subspace Orthogonality Relations

(Row A)^perp = Nul A, and (Col A)^perp = Nul A^T. The orthogonal complement of the row space is the null space of A.