Linear and multilinear algebra; matrix theory Definition

Orthogonal matrix

Definition

Let $n \ge 1$ and let $Q$ be a real $n \times n$ matrix. The matrix $Q$ is orthogonal when $Q^T Q = I_n,$ where $Q^T$ is the transpose and $I_n$ is the identity matrix.

Remarks

Equivalently, $Q$ is invertible with $Q^{-1} = Q^T$ , and then also $Q Q^T = I_n$ . Equivalently, the columns of $Q$ form an orthonormal basis of $\R^n$ ; the same holds for the rows. The map $x \mapsto Qx$ keeps the dot product, so it is a linear isometry of $\R^n$ . The determinant of an orthogonal matrix is $1$ or $-1$ .

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Identity matrix object
Matrix definition
Transpose definition
Orthogonal matrix definition

Tag	`0QRBVFJ50X`
URI	`m2://0QRBVFJ50X`
Kind	definition
Writing	W1 stated
MSC	15B10
Defines	Orthogonal matrix
Uses	Transpose, Identity matrix

Orthogonal matrix

Definition

Remarks

Reading path

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