Can a non-invertible matrix be extended to an invertible one?

Joshua Tilley 06/12/2018. 2 answers, 141 views
linear-algebra matrices linear-transformations

For a square matrix $M$ call any square matrix M' of the form $$\left(\begin{array}{cc} M & A\\ B & C \end{array}\right)$$ an extension of $M$. Does it follow that if $M$ is not invertible that all extensions $M'$ are not invertible? I believe the answer is no. If not, is there an extension that is invertible? Can we prove that there always is?

2 Answers


For any $M$, the matrix $$\pmatrix{M&I\\I&0}$$ is invertible.


Robert Lewis 06/12/2018.

My answer is similar in spirit to Lord Shark the Unknown's, but provides a few more details.

First of all, as Lord Shark affirms, for any matrix square matrix $M$, the extension matrix $E_M$,

$E_M = \begin{bmatrix} M & I \\ I & 0 \end{bmatrix}, \tag 1$

is invertible. The easiest way to see this is to show that

$\ker E_M = \{ 0 \}; \tag 2$

now if

$\text{size}(M) = n, \tag 3$

that is, $M$ is an $n \times n$ matrix over some field $\Bbb F$, then

$\text{size}(E_M) = 2n; \tag 4$

thus $E_M$ may be considered as operating on the $2n$-dimensional vector space $\Bbb F^{2n}$, any vector $v \in \Bbb F^{2n}$ of which may be written in "stacked form"

$v = \begin{pmatrix} x \\ y \end{pmatrix}, \tag 5$

where $x, y \in \Bbb F^n$; then if

$E_M v = 0, \tag 6$

we have

$\begin{pmatrix} Mx + y \\ x \end{pmatrix} = \begin{bmatrix} M & I \\ I & 0 \end{bmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = 0, \tag 7$

from which we conclude

$x = 0, \tag 8$

and

$y = -Mx = -M(0) = 0; \tag 9$

thus

$v = 0, \tag{10}$

which shows that (2) binds and thus that $E_M$ is invertible. We may in fact find the inverse $E_M^{-1}$ by setting

$E_M v = w, \tag{11}$

where

$w = \begin{pmatrix} s \\ t \end{pmatrix}; \tag{12}$

then as above we have

$Mx + y = s, \; x = t, \tag{13}$

whence

$y = s - Mt; \tag{14}$

(13) and (14) together show that

$E_M^{-1} = \begin{bmatrix} 0 & I \\ I & -M \end{bmatrix}, \tag{15}$

which we may easily check:

$E_M^{-1} w = \begin{bmatrix} 0 & I \\ I & -M \end{bmatrix} \begin{pmatrix} s \\ t \end{pmatrix} = \begin{pmatrix} t \\ s - Mt \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix}, \tag{16}$

using (13) and (14).


HighResolutionMusic.com - Download Hi-Res Songs

1 BLACKPINK

Kiss And Make Up flac

BLACKPINK. 2018. Writer: Soke;Kny Factory;Billboard;Chelcee Grimes;Teddy Park;Marc Vincent;Dua Lipa.
2 Martin Garrix

Waiting For Tomorrow flac

Martin Garrix. 2018. Writer: Pierce Fulton;Mike Shinoda;Martijn Garritsen;Brad Delson.
3 Martin Garrix

Access flac

Martin Garrix. 2018. Writer: Martin Garrix.
4 John Legend

Written In The Stars flac

John Legend. 2018. Writer: Kiana Brown;Santoy;Kevin White;Mike Woods;MZMC;The Heavy Group;Rice N' Peas.
5 Martin Garrix

Yottabyte flac

Martin Garrix. 2018. Writer: Martin Garrix.
6 Alan Walker

Diamond Heart flac

Alan Walker. 2018. Writer: Alan Walker;Sophia Somajo;Mood Melodies;James Njie;Thomas Troelsen;Kristoffer Haugan;Edvard Normann;Anders Froen;Gunnar Greve;Yann Bargain;Victor Verpillat;Fredrik Borch Olsen.
7 Cardi B

Taki Taki flac

Cardi B. 2018. Writer: Bava;Juan Vasquez;Vicente Saavedra;Jordan Thorpe;DJ Snake;Ozuna;Cardi B;Selena Gomez.
8 Bradley Cooper

Shallow flac

Bradley Cooper. 2018. Writer: Andrew Wyatt;Anthony Rossomando;Mark Ronson;Lady Gaga.
9 Post Malone

Sunflower flac

Post Malone. 2018. Writer: Louis Bell;Billy Walsh;Carter Lang;Swae Lee;Post Malone.
10 Dyro

Latency flac

Dyro. 2018. Writer: Martin Garrix;Dyro.
11 Zara Larsson

Ruin My Life flac

Zara Larsson. 2018. Writer: Delacey;Michael Pollack;Stefan Johnson;Jordan Johnson;Sermstyle;Jackson Foote.
12 ZAYN

Fingers flac

ZAYN. 2018. Writer: Zayn Malik;Alex Oriet;David Phelan.
13 Halsey

Without Me flac

Halsey. 2018. Writer: Halsey;Delacey;Louis Bell;Amy Allen;Justin Timberlake;Timbaland;Scott Storch.
14 Lady Gaga

I'll Never Love Again flac

Lady Gaga. 2018. Writer: Benjamin Rice;Lady Gaga.
15 Sia

I'm Still Here flac

Sia. 2018. Writer: Sia.
16 Dewain Whitmore

Burn Out flac

Dewain Whitmore. 2018. Writer: Dewain Whitmore;Ilsey Juber;Emilio Behr;Martijn Garritsen.
17 Little Mix

Woman Like Me flac

Little Mix. 2018. Writer: Nicki Minaj;Steve Mac;Ed Sheeran;Jess Glynne.
18 Mako

Rise flac

Mako. 2018. Writer: Riot Music Team;Mako;Justin Tranter.
19 Blinders

Breach (Walk Alone) flac

Blinders. 2018. Writer: Dewain Whitmore;Ilsey Juber;Blinders;Martin Garrix.
20 Rita Ora

Let You Love Me flac

Rita Ora. 2018. Writer: Rita Ora.

Related questions

Hot questions

Language

Popular Tags