Transpose

In linear algebra, the transpose of a matrix A is another matrix A^T (also written A′, A^tr, ^tA or A^t) created by any one of the following equivalent actions:

Formally, the i th row, j th column element of A^T is the j th row, i th column element of A:

If A is an m × n matrix then A^T is an n × m matrix.

The transpose of a matrix was introduced in 1858 by the British mathematician Arthur Cayley.

Examples

Properties

For matrices A, B and scalar c we have the following properties of transpose:

The operation of taking the transpose is an involution (self-inverse).

The transpose respects addition.

Note that the order of the factors reverses. From this one can deduce that a square matrix A is invertible if and only if A^T is invertible, and in this case we have (A⁻¹)^T = (A^T)⁻¹. By induction this result extends to the general case of multiple matrices, where we find that (A₁A₂...A_k−1A_k)^T = A_k^TA_k−1^T...A₂^TA₁^T.

Transposition (chess)

A transposition in chess is a sequence of moves that results in a position which may also be reached by another, more common sequence of moves. Transpositions are particularly common in opening, where a given position may be reached by different sequences of moves. Players sometimes use transpositions deliberately in order to avoid variations they dislike, lure opponents into unfamiliar or uncomfortable territory or simply to worry opponents.

In chess the verb "transpose" means shifting the game on to a different opening track from that on which it started.

Transposition tables are an essential part of a computer chess program.

Examples

Positions reached by different routes

For instance, the first position can be obtained from the Queen's Gambit:

But this position can also be reached from the English Opening:

so the English Opening has transposed into the Queen's Gambit.

The second position shows another example. The position can arise from the French Defence.

Bad Suns

Bad Suns is an American alternative rock band that formed in 2012, and now consists of Christo Bowman, Gavin Bennett, Miles Morris and Ray Libby. A majority of the band members are from Los Angeles, California. The group has been signed to Vagrant Records, where they released their debut album Language & Perspective in 2014. The band's sound is inspired from '70s and '80s post-punk pioneers like The Cure and Elvis Costello.

Background

The Southern California rock band Bad Suns formed in 2012. Made up of Christo Bowman (vocals), Gavin Bennett (bass), Miles Morris (drums) and Ray Libby (guitar), the four piece group ranges from ages 20–23, yet has a sound reminiscent of rock stalwarts from generations past. “I grew up with a lot of world music playing in the house. When I was 10, I started getting heavily interested in the guitar, and my dad began introducing me to his records from the 70’s and the 80’s. Initially Elvis Costello, then to The Clash, The Cure, and so on,” notes Libby, “all of these artists and bands had a big impact on me, at a young age, as far as song composition goes.” The band’s debut EP Transpose features angst-ridden riffs and ethereal vocals reminiscent of post-punk legends of the early 80’s. “I started writing my first songs at that time," Bowman continues, “Though we can now reflect on that era of music, those artists were ahead of their time in a lot of ways. That’s what’s most inspiring.”