True or false: (a) A matrix whose columns form an orthogonal basis of ℝ^n is an orthogonal matri

step 1 So true or false, applying elementary row operations to an orthogonal matrix produces an orthogo

True or false: (a) A matrix whose columns form an orthogonal...

True or false: (a) A matrix whose columns form an orthogonal basis of $\mathbb{R}^n$ is an orthogonal matrix. (b) A matrix whose rows form an orthonormal basis of $\mathbb{R}^n$ is an orthogonal matrix. (c) An orthogonal matrix is symmetric if and only if it is a diagonal matrix.

Key Concepts

Orthogonal Matrix

An orthogonal matrix is a square matrix with orthonormal columns and rows, meaning that each column (or row) has unit length and is perpendicular to all other columns (or rows). This definition implies that the matrix’s transpose is equal to its inverse. Orthogonal matrices preserve lengths and angles in Euclidean space, making them fundamental in rotations, reflections, and other rigid transformations.

Orthonormal Basis

An orthonormal basis of a vector space is a set of vectors that are both orthogonal to each other and of unit length. This concept is central in linear algebra as it simplifies many calculations including coordinate transformations and inner products, and it serves as the foundation for defining orthogonal matrices when these vectors are arranged as columns or rows in a square matrix.

Symmetric Matrix

A symmetric matrix is one that is equal to its transpose, meaning the element in the ith row and jth column is the same as the element in the jth row and ith column. This property ensures that all eigenvalues are real and that the matrix is diagonalizable by an orthogonal transformation, which is particularly useful in optimization and in systems where energy or quadratic forms are involved.

Diagonal Matrix

A diagonal matrix is characterized by having all off-diagonal entries equal to zero, with only the main diagonal possibly containing non-zero elements. Diagonal matrices simplify many aspects of linear algebra, such as computing powers and exponentials of matrices, and in contexts involving symmetric and orthogonal matrices, a symmetric orthogonal matrix that is also diagonal necessarily has its diagonal entries as ±1.

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