For an instant John longed for the sound of the bells of Noyon-la-Sainte, the touch of his mother's hand, the lilt of Charles's voice in the square raftered rooms, his father's bass tones rumbling to the canons, and the sight of the beloved bishop.

For any such square the middle corner of these will be called the vertex of the square and the corner not on the curve will be called the diagonal point of the square.

For the Lo Shu square was a remarkably complete compendium of most of the chief religious and philosophical ideas of its time.

For example, a certain landowner might have been said to own 32, 000 acres of land, not 50 square miles of land.

For example, a vertex configuration of ( 4, 6, 8 ) means that a square, hexagon, and octagon meet at a vertex ( with the order taken to be clockwise around the vertex ).

For example, adjacent to square 5F are squares 4F, 6F, 5E, and 5G, but not 6E or 4G.

For positive values of a and b, the binomial theorem with n = 2 is the geometrically evident fact that a square of side can be cut into a square of side a, a square of side b, and two rectangles with sides a and b. With n = 3, the theorem states that a cube of side can be cut into a cube of side a, a cube of side b, three a × a × b rectangular boxes, and three a × b × b rectangular boxes.

** For example, consider the map ( which is the " realification " of the complex square function ) where U

For example, suppose the word square is defined by " A figure is square if and only if it has four perpendicular sides of equal length.

For example, JPEG compression uses a variant of the Fourier transformation ( discrete cosine transform ) of small square pieces of a digital image.

For example, both the Egyptians and the Babylonians were aware of versions of the Pythagorean theorem about 1500 years before Pythagoras ; the Egyptians had a correct formula for the volume of a frustum of a square pyramid ;

For example, a 4-cycle ( square ) has girth 4.

For houses in and around Geneva, the average price was 11, 595 Swiss francs ( CHF ) per square metre () ( June 2011 ), with a lowest price per square metre () of 4, 874 Swiss francs ( CHF ), and a maximum price of 21, 966 Swiss francs ( CHF ).

For instance, the geometric mean of two numbers, say 2 and 8, is just the square root of their product ; that is.

# For a given orbit, the ratio of the cube of its semi-major axis to the square of its period is constant.

For a number written in scientific notation, this logarithmic rounding scale requires rounding up to the next power of ten when the multiplier is greater than the square root of ten ( about 3. 162 ).

For problems with all four properties, the time for a quantum computer to solve this will be proportional to the square root of the number of inputs.

For other examples, see proof that the square root of 2 is not rational and Cantor's diagonal argument.

For example, the proof that the column rank of a matrix over a field equals its row rank yields for matrices over division rings only that the left column rank equals its right row rank: it does not make sense to speak about the rank of a matrix over a division ring.

For example Sp ( 8 ) has a representation in terms of 4x4 quaternion unitary matrices which has a 16 dimensional real representation and so might be considered as a candidate for a gauge group.

For general matrices, Gaussian elimination is usually considered to be stable in practice if you use partial pivoting as described below, even though there are examples for which it is unstable.

For example, the orthogonal group O < sub > n </ sub >( R ) consists of matrices A with AA < sup > T </ sup > = 1, so the Lie algebra consists of the matrices m with ( 1 + εm )( 1 + εm )< sup > T </ sup > = 1, which is equivalent to m + m < sup > T </ sup > = 0 because ε < sup > 2 </ sup > = 0.

For example, it is not rare to find studies with character matrices based on whole mitochondrial genomes (~ 16, 000 nucleotides, in many animals ).

For example, methods based on covariance matrices are typically employed on the premise that numbers, such as raw scores derived from assessments, are measurements.

For this reason, positive definite matrices play an important role in optimization problems.

For arbitrary square matrices M, N we write M ≥ N if M − N ≥ 0 ; i. e., M − N is positive semi-definite.

For those media in which the amplitudes are unchanged but a differential phase delay occurs, the Jones matrix is unitary, while those affecting amplitude without phase have Hermitian Jones matrices.

For instance, the general linear group GL ( n, R ) of all invertible n-by-n matrices with real entries can be viewed as a topological group with the topology defined by viewing GL ( n, R ) as a subset of Euclidean space R < sup > n × n </ sup >.

For this reason, generalizations of vector operations to matrices ( e. g. in matrix calculus and statistics ) often involve a trace of matrix products.

For this reason, fuzzy logic usually uses IF-THEN rules, or constructs that are equivalent, such as fuzzy associative matrices.

For a linear continuous-time system, like the example above, described by matrices,,, and, the output controllability matrix

For example, quantitative preparative native continuous polyacrylamide gel electrophoresis ( QPNC-PAGE ) is a method for separating native metalloproteins in complex biological matrices.

For example, it is often desirable to compute an orthonormal basis for a space, or an orthogonal change of bases ; both take the form of orthogonal matrices.

For example, the general linear group over R ( the set of real numbers ) is the group of n × n invertible matrices of real numbers, and is denoted by GL < sub > n </ sub >( R ) or GL ( n, R ).

For quaternion scalars and matrices:

For two matrices

For diagonalizable matrices, an even better method is to use the eigenvalue decomposition of A.

For two matrices of the same dimensions, there is the Hadamard product, also known as the entrywise product and the Schur product.

For two matrices A and B of the same dimensions, the Hadamard product A ○ B is a matrix of the same dimensions, which has elements

For two matrices A and B of any different dimensions m × n and p × q respectively ( no contraints on the dimensions of each matrix ), the Kronecker product denoted A ⊗ B is a matrix with dimensions mp × nq, which has elements

* For any positive integer n, the set of all n by n unitary matrices with matrix multiplication forms a group, called the unitary group U ( n ).